Rigidity of Transfers and Unraveling in

Matching Markets∗

Songzi Du† Yair Livne‡

January 10, 2016

Abstract

We study the role of transfers in the timing of matching. In our model, agents

have the option of matching early and exiting in period 1, before new agents arrive

in period 2; in period 2 after all agents arrive a stable matching is implemented for

those who are present. We prove that without flexible transfers, when the number

of agents in period 1 is large and the number of new arrivals is small, on average

at least one quarter of all agents in period 1 have strict incentives to match early, if

they anticipate others are participating in the stable matching of period 2. We define

a minimal notion of sequential stability and prove that without flexible transfers the

probability of sequential stability tends to 0 as the number of period-one agents gets

large. We show that flexible transfers eliminate these timing problems.

∗We are grateful to Mike Ostrovsky, Andy Skrzypacz and Bob Wilson for guidance during research andextensive discussion. And for helpful comments we thank Yossi Feinberg, Alex Frankel, Ben Golub, JohnHatfield, Fuhito Kojima, Scott Kominers, Vijay Krishna, Eli Livne, Paul Milgrom, Muriel Niederle, andMichael Peters. A previous version of this paper was circulated under the title “Unraveling and Chaos inMatching Markets.”†Simon Fraser University, Department of Economics. Email: songzid@sfu.ca‡Quora, Inc., Mountain View, CA. Email: yair@quora.com

1

1 Introduction

In this paper we analyze a dynamic matching model with the possibility of early matching

before all agents (or all information) arrive to the market. We show that flexible transfers

are crucial for preventing early matching. Moreover, we show that without flexible transfers

there may not even exist an intertemporal matching scheme that is stable (even though the

market would be stable if it was static!). Surprisingly, this problem gets particularly acute if

the market is large, counter to our intuitions that thick markets always operate better than

thin ones.

Many two-sided labor matching markets suffer from unraveling and intertemporal insta-

bilities, with matching decisions being taken before the normal timeline or market partici-

pants failing to coordinate on a timeline. Some well-known examples of unraveling include

the markets for federal judicial clerks, clinical psychology interns, and medical residents1

(Roth and Xing, 1994). In particular, in the judicial clerk market there have been six failed

attempts between 1978 and 1998 to coordinate the timing of hiring (Avery, Jolls, Posner, and

Roth, 2007).2 Consistent with our results, many of the markets that suffer from unraveling

are ones where the wages are more or less fixed and not up for negotiation. For the example

of judicial clerk market, the wage schedule is fixed by the US Congress.

In our model agents are either men or women, are risk neutral, and generate surplus

through matching. We compare two regimes of dividing the surplus from matching: in the

fixed-transfer regime, there is an exogenously fixed division of surplus, perhaps due to some

institutional constraint; in the flexible-transfer regime, the division of surplus is endogenously

determined along with the matching. We assume that matching can take place in one of the

two periods: in the first period, some men and women are present and have the option of

matching early (and exiting), or waiting for the second period; in the second period, which

corresponds to the normal timeline of matching, some new men and women arrive, and a

stable matching for the agents present is implemented, perhaps by a centralized institution.

One can interpret the new agents arriving in the second period in many ways. For example,

the new arrivals may be unobserved agents, due to the imperfection of information in the

first period. In student-job matching, the new arrivals can be new job openings on one side,

1Wetz, Seelig, Khoueiry, and Weiserbs (2010) estimate that in 2007 one in five resident positions in primarycare specialties was offered outside of the centralized National Resident Matching Program (NRMP); thedata in Wetz, Seelig, Khoueiry, and Weiserbs (2010) suggests that unraveling still exists in the medicalresident market, reminiscent of the situation before the introduction of NRMP.

2The most recent attempt to coordinate the timing of hiring in the judicial clerk market was initiated in2004 and abandoned in 2013.

2

and students unavailable for early interviews (say, due to incomplete preparation) on the

other side. In the judicial clerk market, the new arrivals may be judges and students who

stick to the regulated timeline of interviewing and hiring for moral reasons.3

Regardless of the interpretation, the uncertainty about the new agents in the second

period creates incentives for unraveling in the fixed-transfer regime: if all other agents were

to wait for the second period, some pairs of agents would have incentives to match early

and leave the market. In our setting, this happens when agents on both sides of the market

are positioned high relative to their surroundings, and thus for these agents waiting for new

arrivals has a larger downside than upside. Our first result states that when the transfers

are fixed and as the market gets large, on average at least one quarter of all agents in

the first period have strict incentives to match early, independent of the underlying type

distributions and surplus function. This is sharply contrasted to the flexible-transfer regime,

which eliminates all incentives to match early. The intuition is simple: when the transfers

are exogenously fixed, they do not reflect the demand-supply condition in period 2 (i.e., they

do not clear the market in period 2); as a result, some agents have incentive to block the

outcome in period 2, and some of those agents may engage in such blocking by matching

early in period 1. In contrast, when the transfers are endogenously determined in period 2,

they accurately reflect the supply-demand condition, so no pair of agents have joint incentive

to block the outcome in period 2, and there is also no joint incentive to block in period 1

(no incentive to match early).

Given our result that a lot of agents have incentives to match early in the fixed-transfer

regime, it is natural to conjecture that agents in the first period would split into two groups,

with one group matching early and the other waiting for the second period; the second

group may be empty, representing the possibility of total unraveling. They could achieve

such a split as an equilibrium outcome or as a result of an organized dynamic matching

mechanism. Avery, Jolls, Posner, and Roth (2007) call this sort of split a “mixed adherence

and nonadherence” to the start date (which corresponds to the second period in our model)

and write about the possibilities of such split being sustained in equilibrium in the judicial

clerks market.

Our second result shows that in large markets with fixed transfers, any mixed adherence

and nonadherence to the start date is difficult to sustain in equilibrium; for most realizations

of types, in every possible early-matching scheme there are agents with incentives to deviate:

3Avery, Jolls, Posner, and Roth (2007) quote judges and students who expressed frustrations over thefact that many interview and offer/accept positions before the regulated timeline.

3

either from waiting for period-two matching to early matching, or vice versa. We use the

term sequential instability to refer to the presence of these deviations. Sequential instability

represents a failure of coordination in the timing of matching and is once again a consequence

of fixed transfers: when the transfers are allowed to freely adjust, no one would have an

incentive to deviate from matching in period 2, and hence everyone matching in period 2 is

sequentially stable.

Finally, we study an intermediate regime between the two extreme points: imagine that

the formal matching institution in period 2 cannot implement flexible transfers due to some

institutional constraint, but if agents match early (i.e., outside of the formal institution)

they may negotiate any desirable level of transfers. We show that flexible transfers in period

1 alone is not sufficient to prevent early matching nor to guarantee sequential stability,

and in general expand the possibilities of early matching. These results thus reinforce the

importance of flexible transfers in the formal matching institution (i.e., period 2 in our

model). Some practical examples of matching institution with flexible transfers include

the proposal of Crawford (2008) for incorporating salaries in National Residence Matching

Program, and of Sonmez and Switzer (2012) and Sonmez (2012) for matching cadets to

military services where the transfers are (additional) years of services. Our paper contributes

a new rationale for the desirability of flexible transfers in matching institutions.4

Our paper compares two classical models of matching: Gale-Shapley (matching without

transfers, or equivalently with fixed transfers) and Becker-Shapley-Shubik (matching with

flexible transfers). It is well-known that the predictions of matching with and without trans-

fers have close analogies in the standard (static) setting: for example, the lattice structure

of stable matchings, the existence of a median stable matching, the incentive compatibility

property of the two extreme stable matchings, etc. (Roth and Sotomayor, 1992). We show

that under a natural dynamic extension matching with and without transfers have drasti-

cally different implications. On highlighting the advantage of flexible transfers, our results

complement Echenique and Galichon (2014), who find that stable matching with fixed trans-

fers can be very inefficient in comparison with stable matching with flexible transfers in the

classical, static model. Relatedly, Jaffe and Kominers (2014) embed the fixed transfers and

flexible transfers models as two extreme points in a model with taxation and study compara-

tive statics on efficiency as transfers become more flexible (i.e., as the percentage of taxation

decreases).

4Previously, Bulow and Levin (2006) have shown that stable matching with fixed wages can lead to wagesuppression (relative to a stable matching with flexible wages) if firms set wages before entering the stablematching mechanism.

4

Our paper complements the classic work on unraveling by Li and Rosen (1998), Suen

(2000) and Li and Suen (2000), who study early matching as a form of insurance due to

risk aversion in settings with flexible transfers; consistent with our results, they show that

early matching cannot be sustained in equilibrium when agents are risk neutral and the

transfers are flexible. Risk aversion is clearly an important reason for early matching and

is applicable whether or not transfers are flexible. We propose a different channel through

which unraveling can arise, and this channel only impacts markets with fixed transfers. Our

work suggests that markets with flexible transfers could be less prone to unraveling.

The literature has come up with other reasons for unraveling in matching markets: over

provision of information (Ostrovsky and Schwarz, 2010), strategic complementarity over the

decision to match early (Echenique and Pereyra, 2014), correlation of matching preferences

(Halaburda, 2010), search costs (Damiano, Hao, and Suen, 2005), exploding offers (Niederle

and Roth, 2009), uncertainty about the imbalance between supply and demand (Niedrele,

Roth, and Unver, 2010), and information flow in social network (Fainmesser, 2013). All of

these papers work in the setting of fixed transfers, and it remains an open question if flexible

transfers could mitigate unraveling through these channels. Sonmez (1999) has shown that in

a many-to-one matching setting with fixed transfers and complete information, unraveling is

inevitable for some realization of preferences. Finally, Roth (1991) empirically demonstrates

that centralized matching mechanisms that are not stable typically leads to unraveling.

The paper proceeds as follows. We present our model in Section 2 and illustrate our

results with simple examples in Section 3. Section 4 shows the prevalence of early matching

incentives and the likelihood of sequential instability when transfers are fixed. Section 5

shows that flexible transfers eliminate these timing problems and examines the intermediate

regime in which flexible transfers are available in period 1, but not in period 2. Section 6

presents some discussion of our results. Most of the proofs are in the Appendix.

2 Model

We study a two-sided, one-to-one matching problem over two periods.

Agents In the first period n men and n women are present in the market. In the second

period, k new men and k new women arrive on the market. Both n and k are positive

integers, and for simplicity are assumed to be common knowledge among the agents.5

5As it will become apparent, our results are unchanged if with probability ε > 0, k men and k womenarrive, and with probability 1− ε no new agents arrive. And likewise if one side of the market has more new

5

Both men and women have types, which are independently and identically drawn (i.i.d.)

from some distributions F and G, respectively. We assume that F is distributed on the

interval [m,m], with a positive and continuous density f on the interior (m,m). Likewise

for distribution G on the interval [w,w] with density g. We assume finite lower bounds m

and w, but the upper bounds m and w can be infinite, i.e., F and G can be distributed over

[m,∞) and [w,∞), respectively. A common example of F and G is the uniform distribution

on [0, 1]; we refer to this distribution as U[0, 1].

For each 1 ≤ i ≤ n we denote by mi the i-th lowest type among the n men present in

the first period. Likewise, women’s types in the first period are denoted by w1 < · · · < wn.

We let m2i be the i-th lowest type man in the second period, and w2

i be the i-th lowest type

woman, 1 ≤ i ≤ l ≤ k + n, where l is the number of men/women present in the second

period; we may have l < n+ k due to early matching and exiting in the first period.

We assume that in the first period the men and women who are present observe each

other’s types. (This assumption is stronger than necessary — see Remark 1.) On the other

hand, agents in the first period do not know the types of the new arrivals in the second

period, although they know the distributions of types.

Utility Agents are risk neural and maximize their expected, quasi-linear utilities. Agents

match to generate a surplus that they then split; the surplus does not depend on the time

(period 1 or 2) of matching. For simplicity, suppose that the utility of an agent who does not

match is zero. We consider two regimes of dividing the surplus between men and women:

1. In the fixed-transfer regime6, when a man of type m is matched to a woman of type

w, man m gets value

U(w | m) ≥ 0 (1)

and woman w gets value

V (m | w) ≥ 0, (2)

where functions U and V represent a fixed scheme of dividing the U(w | m)+V (m | w)

units of surplus produced by man m with woman w.7 For example, U(m | w) can be

the fixed schedule of salary received by the worker (man) from the firm (woman) when

they are matched together, contingent on their types.

arrivals than the other with probability 1/2 each.6We have in mind applications of our model to firm-worker matching, so we emphasize the term “fixed

transfer” instead of the term “no transfer” which is commonly used in the literature.7Mailath, Postlewaite, and Samuelson (2013) call U(w | m) and V (m | w) premuneration values.

6

We assume that the functions U and V are twice continuously differentiable, increasing

in matched type:∂U(w | m)

∂w> 0,

∂V (m | w)

∂m> 0, (3)

and have bounded ratio of derivatives: there exists a positive constant b such that

∂U(w | m)

∂w

/∂U(w′ | m)

∂w≤ b,

∂V (m | w)

∂m

/∂U(m′ | w)

∂m≤ b, (4)

for all m,m′ ∈ [m,m] and w,w′ ∈ [w,w].

2. In the flexible-transfer regime, men and women can freely negotiate the division of

surplus: when man of type m is matched to woman of type w, man m gets

U(w | m) + P (m,w) ≥ 0, (5)

and woman w gets

V (m | w)− P (m,w) ≥ 0, (6)

where the transfer P (m,w), which can be positive or negative, is endogenously de-

termined (i.e., as a part of the stable matching). We assume that the total surplus

U(w | m) + V (m | w) features complementarities between the type of man and the

type of woman:∂2 (U(w | m) + V (m | w))

∂m∂w> 0. (7)

2.1 Fixed-transfer Regime

Second Period We assume that in the second period there is a centralized institution

that implements a stable matching for all agents who are present in the second period —

which consist of first-period agents who have decided to wait and the new arrivals. Let

m2l > · · · > m2

1 and w2l > · · · > w2

1 be the men and women present in the second period.

A matching in the second period is a subset µ2 ⊂ {m21, . . . ,m

2l } × {w2

1, . . . , w2l } such that

every man is matched to a distinct woman.8 The matching µ2 depends on the types in the

second period: µ2 = µ2(m21, . . . ,m

2l , w

21, . . . , w

2l ). To keep the notation simple, we omit this

dependence.

8Formally, this means that (1) for every m2i , there exists a j such that (m2

i , w2j ) ∈ µ2, and (2) if

(m2i , w

2j ), (m2

i , w2j′) ∈ µ2, then j = j′.

7

For the fixed-transfer regime, we use the stable matching of Gale and Shapley (1962):

Definition 1. The matching µ2 of the second period is stable if for every man m2i and every

woman w2j in the second period, we have

U(w2i′ | m2

i ) ≥ U(w2j | m2

i ), or V (m2j′ | w2

j ) ≥ V (m2i | w2

j ), (8)

where (m2i , w

2i′) ∈ µ2 and (m2

j′ , w2j ) ∈ µ2. 9

Condition (8) says that no pair of man and woman has a joint incentive to deviate from

µ2 (no blocking). By the assumption in (3), a man always prefers a woman of higher type

over a woman of lower type, and likewise for the preference of woman. Therefore, a stable

matching here is simply an assortative matching, i.e., the highest type man is matched to

the highest type woman, the second highest to the second highest, etc. So we assume a

stable/assortative matching is always implemented in period 2.

First Period and Early Matching In the first period men and women may match early

and permanently leave the market, gaining utility from their match partner; or they may

wait for the stable matching in the second period. Given men mn > · · · > m1 and women

wn > · · · > w1 in the first period, an early matching µ ⊂ {w1, . . . , wn} × {w1, . . . , wn} is a

set of man-woman pairs such that each agent belongs to at most one pair.10 We emphasize

that unlike the second-period matching µ2 which is a complete matching, the early matching

µ could be a partial matching; in particular, we could have µ = ∅, which represents the

situation where all agents wait to the second period.

We are interested in early matchings from which no pair of agents has an incentive to

deviate. Let µ2 be the stable (i.e., assortative) matching in the second period. Given a list

L = {mi : i ∈ I} ∪ {wi : i ∈ J} of men and women who wait to the second period (where

I, J ⊆ {1, . . . , n} and |I| = |J |), and anticipating the new arrivals and the assortative

matching µ2 in the second period, man mi ∈ L, has expected utility:

U(L,mi) ≡k∑j=0

(k

j

)F (mi)

j(1− F (mi))k−j · Ek[U(w2

r+j | mi) ], (9)

where r is the rank (from the bottom) of mi in {mi′ : i′ ∈ I}, w2r+j is the (r + j)-th

9Given conditions (1) and (2) on utility functions, the individual rationality requirement in stability isautomatically satisfied.

10Formally, we have (mi, wj), (mi, wj′) ∈ µ⇒ j = j′ and (mi′ , wj), (mi, wj) ∈ µ⇒ i = i′.

8

lowest woman in the second period, among the k new women and {wi′ : i′ ∈ J}, and

the expectation Ek is taken over the type realizations of the k new women. In (9), with

probability(kj

)F (mi)

j(1−F (mi))k−j there are j number of new men with types below mi; in

this event man mi is assortatively matched to the woman of rank r+ j in the second period.

Likewise, by waiting to the second period woman wi ∈ L, has expected utility:

V(L,wi) ≡k∑j=0

(k

j

)G(wi)

j(1−G(wi))k−j · Ek[V (m2

r+j | wi) ], (10)

where r is the rank (from the bottom) of wi is in {wi′ : i′ ∈ J}, and m2r+j is the (r + j)-th

lowest man in the second period, among the k new men and {mi′ : i′ ∈ I}.We are interested in early matchings that are stable in conjunction with the second-period

assortative matching:

Definition 2. Fix a realization of types in the first period. The matching scheme (µ, µ2) is

sequentially stable for this realization if:

1. the second-period matching µ2 is stable in the sense of Definition 1; so µ2 is an assor-

tative matching;

2. for any man mi and woman wj who both wait for the second period ((mi, wj′) 6∈ µ for

every j′, and (mi′ , wj) 6∈ µ for every i′), we have

U(L(µ),mi) ≥ U(wj | mi) or V(L(µ), wj) ≥ V (mi | wj), (11)

where L(µ) is the list of men and women who wait for the second period according to

µ;

3. for any couple (mi, wj) who matches early ((mi, wj) ∈ µ), we have both

U(wj | mi) ≥ U(L(µ)∪{mi, wj},mi) and V (mi | wj) ≥ V(L(µ)∪{mi, wj}, wj). (12)

The sequential aspect of Definition 2 is similar to that of the subgame perfect equilibrium

concept: agents anticipate a stable matching µ2 in the second period (Point 1), and the early

matching µ is pairwise stable given this anticipation (Points 2 and 3). Point 2 says that every

pair of agents designated to wait on the market by µ does not want to deviate and match

early (at least one member of the pair does not have incentive to match early). Point 3 says

9

that both members of a pair designated by µ to match early prefer the designated match

over waiting for the second period.

In Definition 2 we have intentionally proposed a minimal notion of sequential stability:

the definition does not require the early matching µ to have no blocking (i.e., does not require

the early matching µ to be assortative). Such minimal definition strengthens our negative

result (Theorem 2) in the fixed-transfer regime.

We sometimes abuse the terminology by saying that an early matching µ is sequentially

stable, which means that the early matching µ, in conjunction with an assortative matching

in the second period, forms a sequentially stable matching scheme in the sense of Definition 2.

Remark 1. We have assumed full observability of the types in period 1 because we want,

as a benchmark, an unambiguous distinction between the known types in period 1 and the

unknown types from the later period. We note, however, that the full observability of the

first-period types is not strictly necessary for agents’ early matching decision. When thinking

about his expected utility from period 2, an agent in period 1 needs not know the types of

those who have matched early. And among those from period 1 who wait, an agent needs not

know the types of those who are clearly out of his reach, i.e., those whose rank differs by more

than k from him. When k is small compared to n, which is assumed for our fixed-transfer

results, this amounts to assuming an agent knowing the types in his local surrounding in

period 1.

2.2 Flexible-transfer Regime

Second Period In the second period of the flexible-transfer regime, when man m2i is

matched to woman w2j (i.e., (m2

i , w2j ) ∈ µ2), they negotiate a transfer P 2(m2

i , w2j ) (which

could be positive or negative), from which man m2i gets

U(w2j | m2

i ) + P 2(m2i , w

2j ) ≥ 0, (13)

and woman w2j gets

V (m2i | w2

j )− P 2(m2i , w

2j ) ≥ 0. (14)

The transfer P 2(m2i , w

2j ) specifies a division of the surplus U(w2

j | m2i ) + V (m2

i | w2j ) created

between man m2i and woman w2

j . We require the payoffs of man and woman to be greater

than or equal to zero, which is a normalization of their outside options. In general, the

transfer P 2(m2i , w

2j ) depend on all types present in the second period, i.e., we abuse the

10

notation by writing

P 2(m2i , w

2j ) = P 2(m2

i , w2j )(m

21, . . . ,m

2l , w

21, . . . , w

2l )

where m21 < m2

2 < · · · < m2l and w2

1 < w22 < · · · < w2

l are the types of all agents present in

the second period.

As in the fixed-transfer regime, we assume that a stable matching is implemented for the

agents present in the second period. In the flexible-transfer regime the stable matching is

defined by Shapley and Shubik (1971) and Becker (1973) and specifies transfers as a part of

the matching:

Definition 3. The matching and transfers (µ2, P 2) of the second period is stable if the non-

negativity conditions (13) and (14) hold for every matched pair, and for every man m2i and

every woman w2j in the second period, we have

(U(w2

i′ | m2i ) + P 2(m2

i , w2i′))

+(V (m2

j′ | w2j )− P 2(m2

j′ , w2j ))≥ U(w2

j | m2i ) + V (m2

i | w2j ),

(15)

where (m2i , w

2i′) ∈ µ2 and (m2

j′ , w2j ) ∈ µ2.

Condition (15) is the analogue of condition (8) in the flexible-transfer regime and says

that no pair of man and woman can find a division of their surplus that both prefer over

what they get in (µ2, P 2) (no blocking). We collectively refer to the matching and transfers

(µ2, P 2) in Definition 3 as a stable matching when there is no possibility of confusion.

Shapley and Shubik (1971) and Becker (1973) have proved that in the flexible-transfer

regime a stable matching (µ2, P 2) always exists, and moreover given complementarities in

the surplus function (Assumption (7)), in any stable matching µ2 must be the assortative

matching. So as in the fixed-transfer regime we assume that an assortative matching is always

implemented in the second period. For every realization of types in the second period, we fix

a schedule of transfers P 2 that make the assortative matching stable. For example, P 2 can be

the transfers that make the stable matching woman-optimal, which can be implemented by

an ascending-price auction (Demange, Gale, and Sotomayor, 1986). Another natural choice

is the transfers associated with the median stable matching (Schwarz and Yenmez, 2011).

First Period and Early Matching As in the fixed-transfer regime, in the first period

agents can match early and skip the stable matching in the second period, which motivates

the definition of sequential stability. Let (µ2, P 2) be the stable matching and transfers in the

11

second period, which means that µ2 is an assortative matching. Given a list L = {mi : i ∈I}∪ {wi : i ∈ J} of men and women who wait to the second period (where I, J ⊆ {1, . . . , n}and |I| = |J |), and anticipating the new arrivals and (µ2, P 2) in the second period, man

mi ∈ L has expected utility:

U(L,mi;P2) ≡

k∑j=0

(k

j

)F (mi)

j(1− F (mi))k−j · Ek[U(w2

r+j | mi) + P 2(mi, w2r+j) ], (16)

where r is the rank (from the bottom) of mi in {mi′ : i′ ∈ I}, w2r+j is the (r + j)-th lowest

woman in the second period, among the k new women and {wi′ : i′ ∈ J}, and the expectation

Ek is taken over the type realizations of the k new women.

Likewise, by waiting to the second period woman wi ∈ L has expected utility:

V(L,wi;P2) ≡

k∑j=0

(k

j

)G(wi)

j(1−G(wi))k−j · Ek[V (m2

r+j | wi)− P (m2r+j, wi) ], (17)

where r is the rank (from the bottom) of wi is in {wi′ : i′ ∈ J}, and m2r+j is the (r + j)-th

lowest man in the second period, among the k new men and {mi′ : i′ ∈ I}.Given these expected utilities in period 1, we are led to the following definition of se-

quential stability:

Definition 4. Fix a realization of types in the first period. The matching scheme (µ, µ2, P 2)

is sequentially stable for this realization if:

1. the second-period matching (µ2, P 2) is stable in the sense of Definition 3; so µ2 is an

assortative matching;

2. for any man mi and woman wj who both wait for the second period ((mi, wj′) 6∈ µ for

every j′, and (mi′ , wj) 6∈ µ for every i′), we have

U(L(µ),mi;P2) + V(L(µ), wj;P

2) ≥ U(wj | mi) + V (mi | wj), (18)

where L(µ) is the list of men and women who wait for the second period according to

µ;

12

3. for any couple (mi, wj) who matches early ((mi, wj) ∈ µ), we have

U(wj | mi) + V (mi | wj) ≥ U(L(µ) ∪ {mi, wj},mi;P2) + V(L(µ) ∪ {mi, wj}, wj;P 2).

(19)

Definition 4 is the analogue of Definition 2 in the flexible-transfer regime: agents antic-

ipates a stable matching (µ2, P 2) in the second period (Point 1), and the early matching

µ is pairwise stable given this anticipation (Points 2 and 3). Point 2 says that for any

pair of agents designated to wait on the market by µ and for any division of their surplus

(U(wj | mi) + V (mi | wj)) from matching early, at least one member of the pair prefers

his/her expected utility from not matching early over his/her division from matching early.

Point 3 says that members of a pair designated by µ to match early can find a division of

their surplus of which both prefer over their expected utilities from not matching early.

As in the fixed transfer regime, Definition 4 is a weak notion of sequential stability since

it does not require the early matching µ to feature no blocking. This weak definition is

irrelevant for the first part of our main result Theorem 3 (since no one matches early), and it

strengthens the claim in the second part that no one matching early is the unique sequentially

stable outcome when the second period matching is woman-optimal (or man-optimal).

3 Examples

In this section we illustrate with examples the incentive to match early (Inequality (12)) as

well as the possible failure of sequential stability (Definition 2) in the fixed-transfer regime.

We then show in the examples how flexible transfers can address these problems.

For the examples in this section, we assume either n = 1 or 2 pairs of agents in period 1,

and k = 1 pair of new arrivals in period 2. Moreover, we assume a uniform distribution of

types: F = G = U[0, 1], and multiplicative value functions U(w | m) = V (m | w) = mw/2.

Clearly, in the fixed-transfer regime these value functions are equivalent to the man getting

a value of w and the woman getting a value of m when man m is matched to woman w.

3.1 Example 1: early matching

We first examine the case of n = 1 and k = 1 in the fixed-transfer regime. Let m ∈ [0, 1] be

the type of the man in period 1, and let w ∈ [0, 1] be the type of the woman in period 1.

13

The incumbent man m prefers to leave the market in the first period with the incum-

bent woman w if his expected match in the second period (from the anticipated assortative

matching) is lower than w. The second-period assortative matching can fall into three cases:

1. if both entrants are of higher type than the incumbents, or if both entrants are of lower

type than the incumbents, then the incumbents are matched in the second period;

2. if the entrant man is of a higher type than m while the entrant woman is of lower

type than w, then the incumbent man is matched to the entrant woman, and gets a

match of quality lower than w. These are the realizations in which the man loses from

waiting, and the woman gains;

3. if the entrant man is of a lower type than m while the entrant woman is of higher type

than w, then the incumbent man is matched to the entrant woman, and gets a match

of quality higher than w. These are the realizations in which the man gains by waiting,

and the woman loses.

Formally, man m’s expected utility from waiting to the second period is (after normalizing

U(w | m) = 12mw to w):

(mw + (1−m)(1− w)

)w + (1−m)

∫ w

0

x dx+m

∫ 1

w

x dx

where the three terms correspond to the three cases above. Comparing this to w and sim-

plifying, man m strictly prefers to match early with w if and only if:11

(1−m)(w)2 > m(1− w)2. (20)

Symmetrically, woman w strictly prefers to match early with m if and only if:

(1− w)(m)2 > w(1−m)2. (21)

Therefore, pairs (m,w) which satisfy both of the above conditions would rather match

early and leave the market over waiting for the assortative matching in the second period.

The set of such pairs is non-empty, and is plotted in Figure 1.

The figure reveals that with a non-negligible probability (approximately 9.7%) the first-

period agents have strict incentive to match early. Agents that prefer an early match (in the

11We write (w)2 for the square of w because we use w2 for the type of woman in period 2.

14

0.2 0.4 0.6 0.8 1.0m

0.2

0.4

0.6

0.8

1.0

w

Figure 1: Types in the shaded area prefer to match early (both (20) and (21) are satisfied).

shaded region) have two notable characteristics: they are of similar type and have relatively

high types. Intuitively, pairs with similar types have a similar probability of gaining or losing

from waiting to period 2. However, since they are of high type and hence the upside from

waiting is smaller than the possible downside. Thus, these agents experience an “endogenous

discounting,” preferring to leave the market. Other agents prefer to wait — if the two types

are not similar, the higher of the two knows that he/she can expect with high probability a

better draw in the second period, and thus prefers to wait. Alternatively, if the two types

are similar but both low, then both agents have a higher upside than downside from waiting,

since a good draw is expected to improve their match by more than a bad one would.

Summarizing, for pairs in shaded region of Figure 1, early matching µ = {(m,w)},in conjunction with the second period assortative matching, forms a sequentially stable

matching scheme (Definition 2). For pairs outside of the shaded region in Figure 1, not

matching early µ = ∅ is sequentially stable.

3.2 Example 2: failure of sequential stability

We now assume n = 2 (and k = 1 as before) in the fixed-transfer regime. Let m2 > m1 be

the types of the two men in the first period, and let w2 > w1 be the types of the two women.

15

The expected utility of man mi in the second period, 1 ≤ i ≤ 2, is (after normalizing

U(w | m) = 12mw to w):(

miwi+(1−mi)(1−wi))wi+(1−mi)

((w−)2 +

∫ wi

w−

x dx

)+mi

((1− w+)w+ +

∫ w+

wi

x dx

),

(22)

assuming that first-period women w− and w+ (w− < wi < w+) are also present in the second

period (if wi is the highest among women waiting for the second period, then let w+ = 1;

likewise for w− = 0). For example, if i = 2, then we always have w+ = 1, with w− = w1 when

w1 waits for period 2, and with w− = 0 when w1 matches early (which reduces to Example

1). The first term in (22) represents the events in which mi is assortatively matched to wi in

the second period, the second term represents events in which mi is assortatively matched

to a worse type than wi, and the last term represents the events in which mi is assortatively

matched to a better type than wi.

It is simple to show (this is a special case of the integration-by-part derivation in Page

47) that mi strictly prefers to match early with wi, i.e., wi strictly dominates (22), if and

only if:

(1−mi)((wi)2 − (w−)2) > mi((1− wi)2 − (1− w+)2), (23)

where the left-hand side represents mi’s downside (being matched downward) in the second

period, and the right-hand side represents his upside (being matched upward). Consistent

with our intuition, the downside increases with the difference between wi and w−, and the

upside increases with the difference between w+ and wi.

Symmetrically, wi prefers to match early with mi if and only if:

(1− wi)((mi)2 − (m−)2) > wi((1−mi)

2 − (1−m+)2), (24)

where m− and m+ are the types of men waiting to the second period who are just below

and above mi.

Let us specialize to the following realization of types in the first period and consider four

cases:

m1 = w1 =2

5, m2 = w2 =

3

5.

1. Consider the option of the top pair (m2, w2) waiting for the second period. If this is

the case, for the incentives of the bottom pair we have: m1 = w1 = 2/5, w− = 0 and

w+ = w2 = 3/5, so inequality (23) holds, i.e., the bottom man’s downside in the second

period dominates his upside, and likewise for the woman. Therefore, the bottom pair

16

has a strict incentive to match early. Intuitively, the presence of the top pair in the

second period imposes strong competition for the bottom pair and caps their upside:

they can get at most 3/5 in the assortative matching of the second period.

2. If the bottom pair indeed matches early, for the incentives of the top pair we have

m2 = w2 = 3/5, w− = 0 and w+ = 1, so again inequality (23) holds, i.e., the top man’s

downside in the second period dominates his upside, and likewise for the woman.

Therefore, the top pair also wants to match early. An alternative way to see this is

to return to Example 1, with n = k = 1, and notice that the pair (3/5, 3/5) is in the

shaded, early-matching region of Figure 1.

3. If indeed the top pair matches early, for the incentives of the bottom pair we have

m1 = w1 = 2/5, w− = 0 and w+ = 1, so now inequality (23) does not hold, i.e, the

bottom man’s upside in the second period now dominates his downside, and likewise

for the woman. Therefore, the bottom pair now wants to wait to the second period.

And indeed in Figure 1 of Example 1, we see that the pair (2/5, 2/5) is not in the

shaded, early-matching region.

4. Finally, if the bottom pair waits for the second period, for the incentives of the top

pair we have m2 = w2 = 3/5, w− = w1 = 2/5 and w+ = 1, so inequality (23) does

not hold, i.e., the top man’s upside in the second period dominates his downside, and

likewise for the woman. Therefore, the top pair also prefers to wait to the second

period. Intuitively, the bottom pair’s presence in the second period provides insurance

(fallback options) for the top pair: they are guaranteed of at least 2/5 in the second

period.

top pair

matches early

bottom pair

matches early

top pair

waits

bottom pair

waits

top pair

waits

bottom pair

matches early

top pair

matches early

bottom pair

waits

Figure 2: Failure of sequential stability.

17

As displayed in Figure 2, where the first arrow corresponds to case 1, etc., each choice

of one pair implies another by the other pair, but none of these choices is consistent with

each other. In other words, no assortative early matching is sequentially stable in the sense

of Definition 2. To complete the argument, we must also consider early matching of cross

matches between higher-ranked and lower-ranked agents. It is easy to see that an early

matching consisting of a single pair of cross match (say, µ = {(m1, w2)}) is never profitable

to the higher-ranked agent in the cross match (i.e., woman w2, because she can do no worse

than m1 in the second period). Furthermore, the early matching of two cross matches

µ = {(m1, w2), (m2, w1)} is not sequentially stable: one can check that in this example if the

early matching µ = {(m1, w2), (m2, w1)} were sequentially stable, then the early matching

µ = {(m1, w1), (m2, w2)} would be sequentially stable as well12 (it is not, as shown by Case

3 in page 17).

Therefore, realization of m1 = w1 = 2/5,m2 = w2 = 3/5 does not admit a sequentially

stable matching scheme.

3.3 Example 3: flexible transfers

Example 1 and 2 are set in the fixed-transfer regime. We now demonstrate in the context of

these examples how flexible transfers can eliminate the incentives to match early and restore

sequential stability.

In the flexible-transfer regime, an agent’s transfer is consistent with his outside options,

i.e., what he can get when he deviates from the assortative matching. For simplicity, we

assume that the woman-optimal stable matching is implemented in period 2; see Example 4

on the role of the woman-optimal stable matching.

Let us first work in the setting of Example 1: n = k = 1. Consider first the case where

the two first-period agents wait to the second period, and let m22 > m2

1 and w22 > w2

1 be the

types in the second period. Recall that in this example the total surplus created by man m

and woman w is U(w | m) + V (m | w) = mw. Then the woman optimal stable matching in

period 2 is the assortative matching of (m2i , w

2i ), i ∈ {1, 2}, together with man m2

1 getting

π1 = 0 (25)

12If m2 has an incentive to match early with w1, then m1 must also have an incentive to match early withw1, since m2 is in a strictly better position than m1 (formally, let wi = w1, w− = 0 and w+ = 1, if (23)holds for mi = m2, then it also holds for mi = m1). Moreover, if m2 has incentive to match early with w1,then he must also have incentive to match early with a better woman w2. And likewise for the women.

18

(after netting his transfer), man m22 getting

π2 = (m22 −m2

1) · w21, (26)

and woman w2i getting

Πi = m2i · w2

i − πi, (27)

for i ∈ {1, 2}. Intuitively, in the woman optimal stable matching each man m2i gets minimally

his outside option which is matching with woman w2i−1 and displacing man m2

i−1. Thus, in

(26) man m22’s outside option is to match with woman w2

1, producing m22 · w2

1 and getting

m22 · w2

1 − Π1 (leaving Π1 to w21 to ensure that she would agree to displace man m2

1).

By construction, we have the following “no blocking” condition:

πi + Πj ≥ m2i · w2

j , (28)

with a strict inequality when i = 1 and j = 2. 13

Inequality (28) eliminates the early matching incentives identified in Example 1 and 2.

To see this, let m and w denote the types of man and woman present in period 1, and

suppose that they wait to period 2, so m = m2i and w = m2

j , where the indices i, j ∈ {1, 2}depend on the realization of the new arrivals’ types. Therefore, man m and woman w have

a joint incentive to match early if and only if

m · w ≥ E[πi + Πj], (29)

where the expectation is taken over the new arrivals’ types (which determine the values of

πi and Πj). By refusing to match early man m gets an expected payoff of E[πi] in period 2,

and woman w an expected payoff of E[Πj]. Therefore, if (29) holds, there exists an division

of m ·w of which both m and w prefer over their expected payoffs; clearly, the converse also

holds. Inequality (29) is the analogue of inequalities (20) and (21) in the flexible-transfer

regime.

However, inequality (28) implies that inequality (29), the incentive to match early, cannot

hold if m < 1 and w > 0: for every realization of types in period 2 inequality (28) implies

that πi+Πj ≥ m2i ·w2

j = m ·w, with a strict inequality when i = 1 and j = 2 (which happens

with a positive probability when m < 1 and w > 0). Therefore, man m and woman w

13By construction, we have π2 +Π1 = m22 ·w2

1. This implies that π1 +Π2 = 0+(m22 ·w2

2−(m22−m2

1) ·w21) >

m21 · w2

2.

19

cannot have a joint incentive to match early. Intuitively, the flexibility of transfers mitigate

the undesirable (to the high type) event of a high type agent being matched to a low type

in period 2 by permitting the high type to extract from the low type a large share of their

joint surplus in such event.

Clearly, the previous analysis extends to Example 2 with n = 2 and k = 1, which shows

that flexible transfers restore sequential stability: since nobody has an incentive to match

early, the outcome that everyone participates in period 2 is sequentially stable.

4 Fixed Transfers

4.1 Prevalence of early matching incentives

Example 1 reveals that in the fixed-transfer regime there is a strictly positive probability of

types with incentive to match early and skip the assortative matching of the second period.

In this subsection we analyze the prevalence of these early matching incentives as the number

of agents gets large. As a measure of prevalence we fix the benchmark of all first-period agents

waiting for the second period and count how many of them have an incentive to deviate from

this benchmark by matching early.

Formally, fix the empty set of early matching µ = ∅ in Definition 2 of sequential stability,

and let ξn be the number of first-period pairs (mi, wi) for whom the incentive to not match

early (Inequality (11) in Definition 2) is violated:

U(L,mi) < U(wi | mi) and V(L,wi) < V (mi | wi),

where L is the set of all agents in the first period, and U(L,mi) and V(L,wi) are, respectively,

mi and wi’s expected utilities from the second period (see Equation (9) and (10)). That is,

ξn =∑n

i=1 1{U(L,mi)<U(wi|mi) and V(L,wi)<V (mi|wi)}.

Note that when counting ξn we restrict to man and woman of the same rank in the first

period. Therefore, ξn is a lower bound on all pairs of man and woman with an incentive to

match early.

Theorem 1. Fix k ≥ 1 pairs of new arrivals. In the fixed-transfer regime, E[ξn]/n tends to

1/4 as the number n of first-period agents tends to infinity.

Proof. See Section A.2.

20

Theorem 1 says that with fixed transfers a lot of first-period agents have incentives to

deviate from the stable matching in period 2, so the empty early matching µ = ∅ is far from

being sequentially stable. The intuition for Theorem 1 is simple. Consider a pair (mi, wi) in

the first period — because of the new arrivals, mi could either do better or do worse than

wi in the assortative matching of the second period. Because mi and wi are of the same

percentile in the distribution, the probability that mi would do better in the second period

is approximately the probability that he would do worse. Therefore, mi has an incentive

to match early with wi if and only if wi is closer to the higher end of her surrounding

(women that mi could be matched to when he does better) than to the lower end (women

that mi could be matched to when he does worse); in large markets, this event happens

with probability 1/2. A symmetric argument works for wi: with probability 1/2 she has an

incentive to match early with mi. Therefore, the ex-ante probability is 1/4 = 1/2× 1/2 that

both mi and wi have incentives to match early. We then reinterpret the ex-ante probability

of 1/4 as the expected fraction of man-woman pairs with an incentive to match early.

4.2 Failure of sequential stability

Theorem 1 implies that in the fixed-transfer regime all first-period agents going to the as-

sortative matching in the second period cannot be sequentially stable. It leaves open the

possibility that some other arrangement of early matching would be sequentially stable. We

now show that this is unlikely in large markets:

Theorem 2. Assume a single pair k = 1 of new arrivals. Then in the fixed-transfer regime,

the probability that the first-period types do not admit a sequentially stable matching scheme

tends to 1 as the number n of first-period agents tends to infinity.

Proof. See Section 4.2.2 and Section A.3.

Theorem 2 says that with fixed transfers and as the market gets large, with probability

tending to 1 any early-matching arrangement (including no early matching) is not sequen-

tially stable: anticipating an assortative matching in the second period, either an individual

man or woman would have an incentive to deviate from his/her early matching by wait-

ing for the second period, or a pair would have incentives to jointly deviate from waiting

for the second period by matching early. The probability of sequential stability vanishes

with the market size n, despite the fact that the number of possible early matchings grows

exponentially with n.

21

The instability in Theorem 2 is driven by the interdependence in the first-period agents’

decisions to match early or to wait for the second period. When the higher-ranked agents

wait they act as strong competitors to the lower-ranked agents, which caps the upside from

waiting (being matched upward in period 2) for the lower-ranked agents, thus making the

lower-ranked agents less willing to wait. On the other hand, when the lower-ranked agents

wait they provide insurance (fallback options) against negative outcome (being matched

downward in period 2) for the higher-ranked agents, thus making the higher-ranked agents

more willing to wait. Thus, the higher-ranked and the lower-ranked agents essentially engage

in a “game” of matching pennies, in which the higher-ranked agents want to synchronize

their time of matching with the lower-ranked agents, while the lower-ranked agents do not

want such synchronization.14 As the number of agents increases these interdependencies

become difficult to reconcile. Therefore, a seemingly minor informational perturbation in

the form of one pair of arriving agents in the second period has a large rippling effect on

the intertemporal stability of the matching market, especially when the market is thick.

Obviously, with more pairs of arrivals the first-period agents care even more about being

matched up or down in period 2, so we expect the same effect to be present; this is indeed

confirmed by Monte Carlo simulations of the next section.

4.2.1 Monte Carlo simulation

We use Monte Carlo simulations to understand how quickly the probability of sequential

stability converge to 0 with n (the number of existing men/women) when k = 1 (the number

of arrivals). Moreover, simulations can tell us what happens when agents have large uncer-

tainty about the second period, i.e., when k is on the same magnitude as n, an interesting

case for which we currently do not have any analytical result. For simplicity, we consider

uniform distribution of types (on the interval [0, 1]) F = G = U[0, 1], and multiplicative

value function U(w | m) = V (m | w) = mw/2; this gives closed-form expressions for agents’

expected utility from period 2 (see Appendix C).15 We plot the results of the Monte Carlo

simulation in Figure 3. For k = 1, the probability of sequential stability rapidly decreases to

32% when n = 20. When k = n, the probabilities of sequential stability are slightly larger

and also decrease with n and k; when n = k = 11 the probability of sequential stability is

around 84%. Figure 3 shows that Theorem 2 is not merely a result in “asymptopia” and

that sequential stability fails with a non-trivial probability even when n is small. Moreover,

14See Example 2 in Section 3.2 for a concrete illustration of this “game” of matching pennies.15The Monte Carlo simulation code is available at http://www.sfu.ca/~songzid/probstablemc.nb.

22

Figure 3 suggests that sequential instability persists when k = n.

æ ææ

ææ

æ

æ

æ

ææ

æ

æ

æ

à à à àà

àà à

àà

1 5 10 15 20n0.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.

Probability of Sequential Stabilityà k � n

æ k � 1

Figure 3: Monte Carlo computation of the probability that a sequentially stable matchingscheme exists in the fixed-transfer regime, given F = U[0, 1] = G and U(w | m) = mw/2 =V (m | w).

4.2.2 Main Step in the Proof of Theorem 2

To illustrate why Theorem 2 is true, we now sketch the main step of the proof in which

we study sequential stability in a related and simpler model; let us call this model the

exponential model.

In the exponential model with parameter r, there are r + 1 men and r + 1 women in

period 1, with types mr > mr−1 > · · · > m0 and wr > wr−1 > · · · > w0. We assume that

the differences in consecutive types, mi+1 − mi and wi+1 − wi, 0 ≤ i ≤ r − 1, are i.i.d.

exponential random variables (with mean 1). Moreover, we assume that agents mr, wr, m0

and w0 always wait for period 2, and we study the early matching decisions of agents mi

and wi, 1 ≤ i ≤ r − 1. We restrict attention to early matching that is assortative: any

assortative early matching can be represented by a subset I ⊆ {1, 2, . . . , r− 1} that lists the

23

indices of agents who choose not to match early. Finally, we assume that the early matching

incentives are given by the following definition of sequential stability:

Definition 5 (Sequential Stability in the Exponential Model). Fix a realization of mi and

wi, 0 ≤ i ≤ r. The early matching in which a subset I = {il : 1 ≤ l ≤ L} wait for period 2,

where i0 ≡ 0 < i1 < · · · < iL < r ≡ iL+1, is sequentially stable for this realization of types if:

1. for any i ∈ I (there exists a unique l such that i = il), either woman wi has no incentive

to match early with man mi in the sense of

mil+1−mi︸ ︷︷ ︸

upside of waiting

≥ mi −mil−1︸ ︷︷ ︸downside of waiting

, (30)

or man mi has no incentive to match early with woman wi in the sense of

wil+1− wi︸ ︷︷ ︸

upside of waiting

≥ wi − wil−1︸ ︷︷ ︸downside of waiting

; (31)

2. for any i 6∈ I (there exists a unique l such that il < i < il+1), the pair (mi, wi) both

have incentives to match early with each other:

mil+1−mi︸ ︷︷ ︸

upside of waiting

≤ mi −mil︸ ︷︷ ︸downside of waiting

, and wil+1− wi︸ ︷︷ ︸

upside of waiting

≤ wi − wil︸ ︷︷ ︸downside of waiting

. (32)

To understand Definition 5, consider the first point, which says that man mil has an

incentive to match early with woman wil if and only if the downside wil −wil−1from waiting

for period 2 exceeds the upside wil+1− wil from waiting. The expression of upside and

downside as the difference in types comes from the assumptions of k = 1 and of a large

market of which the agents m0, . . . ,mr, w0, . . . , wr are a part, so with large probability mil

is either matched to wil+1(when the new man is worse than mil and the new woman is

better than wil+1) or to wil−1

(when the new man is better than mil and the new woman

is better than wil−1). (We formalize this point in Section A.3.) Since when deciding about

early matching the agents simply compare the differences in types in their surrounding, in

the exponential model it is without loss of generality to restrict to assortative, sequentially

stable early matching: any non-assortative early matching that is sequentially stable can be

“uncrossed” and converted into an assortative early matching that is also sequentially stable

(types in their surrounding are not affected by the uncrossing). Finally in a large market,

24

the differences of order statistics mi+1−mi and wi+1−wi, 0 ≤ i ≤ r− 1, can be normalized

to be i.i.d. exponential (Lemma 4); see also Pyke (1965).

Definition 5 critically assumes that agents m0, mr, w0 and wr exogenously wait for period

2. In Section A.3 we justify this assumption by proving that in large markets it is unlikely

to have a large number of consecutively ranked pairs all choosing to match early, so in any

sequentially stable matching scheme agents who wait (candidates for m0, mr, w0 and wr)

can be found everywhere in the support of type.

Let P̂r be the probability measure for the random variables mi and wi with exponentially

distributed differences. For any I ⊆ {1, 2, . . . , r − 1}, let GI denote the event (i.e., a set of

realizations of mi and wi) that the early matching implied by I is sequentially stable in the

sense of Definition 5; see Figure 4 for an example of GI . We now show the probability of a

sequentially stable I tends to zero with r in the exponential model:

limr→∞

P̂r

⋃I⊆{1,2,...,r−1}

GI

= 0. (33)

We first exploit the memoryless property of the exponential distribution to simplify the

expression for each individual P̂r(GI):

Lemma 1. Suppose that I = {il : 1 ≤ l ≤ L}, where i0 ≡ 0 < i1 < · · · < iL < r ≡ iL+1.

Then we have:

P̂r(GI) =1

4r−L−1P̂r

(L⋂l=1

({mil+1

−mil ≥ mil −mil−1} ∪ {wil+1

− wil ≥ wil − wil−1}))

.

(34)

Proof. In Point 2 of Definition 5, if il+1 > 2+ii, then we only need to check the early matching

incentives for the pair (m1+il , w1+il), since if (m1+il , w1+il) have incentives to match early,

then so do (m2+il , w2+il). Thus, we have

P̂r(GI) = P̂r

((L⋂l=0

{mil+1−m1+il ≤ m1+il −mil} ∩ {wil+1

− w1+il ≤ w1+il − wil}

)⋂(

L⋂l=1

({mil+1

−mil ≥ mil −mil−1} ∪ {wil+1

− wil ≥ wil − wil−1})))

.

The exponential distribution has no “memory,” that is, conditional on the event mil+1−

m1+il ≤ m1+il − mil , the random variable mil+1− mil has the same distribution as its

25

Figure 4: An example of early matching (indicated by the grey dashed lines) in the ex-ponential model with r = 4. Those who wait for period 2 have indices in I = {1}. GI ={m4−m2 ≤ m2−m1}∩{w4−w2 ≤ w2−w1}∩({m4−m1 ≥ m1−m0}∪{w4−w1 ≥ w1−w0}).Lemma 1 says that P̂r(GI) = P̂r({m4 −m1 ≥ m1 −m0} ∪ {w4 − w1 ≥ w1 − w0})/16.

m4

m3

m2

m1

m0

w4

w3

w2

w1

w0

unconditional distribution,16 and likewise for wi’s. Therefore, we have

P̂r(GI) =L∏l=0

P̂r({mil+1

−m1+il ≤ m1+il −mil})P̂r({wil+1

− w1+il ≤ w1+il − wil})

· P̂r

(L⋂l=1

({mil+1

−mil ≥ mil −mil−1} ∪ {wil+1

− wil ≥ wil − wil−1}))

=1

4r−L−1· P̂r

(L⋂l=1

({mil+1

−mil ≥ mil −mil−1} ∪ {wil+1

− wil ≥ wil − wil−1}))

,

16Suppose that a1, a2, . . . , ai are i.i.d. exponential random variables. Conditioning on the event thata1 ≥ a2 + . . . + an, the random variable

∑nj=1 aj has conditional density function h(z), where (note that

P(a1 ≥ a2 + . . .+ an) = 1/2n−1):

h(z) =1

1/2n−1

∫ z

a1=z/2

exp(−a1)(z − a1)n−2 exp(−(z − a1))

(n− 2)!da1 =

zn−1 exp(−z)(n− 1)!

which is the unconditional density function of∑n

j=1 aj .

26

where the second line follows from Lemma 5 which implies that

P̂r({mil+1

−m1+il ≤ m1+il −mil})

= P̂r({wil+1

− w1+il ≤ w1+il − wil})

=1

2il+1−il−1.

In light of Lemma 1, for any r ≥ 2 and any I = {il : 1 ≤ l ≤ L}, where i0 ≡ 0 < i1 <

· · · < iL < r ≡ iL+1, we define

πr(I) ≡ 1

4r−L−1P̂r

(L⋂l=1

({mil+1

−mil ≥ mil −mil−1} ∪ {wil+1

− wil ≥ wil − wil−1}))

(35)

and

πr ≡∑

I⊆{1,2,...,r−1}

πr(I). (36)

Lemma 2. For any I 6= I ′ ⊆ {1, . . . , r − 1}, we have P̂r(GI ∩GI′) = 0. Hence we have

P̂r

⋃I⊆{1,2,...,r−1}

GI

= πr. (37)

The proof of Lemma 2 is a bit involved so we defer it to the appendix (page 40).

Proposition 1. πr tends to 0 as r →∞.

Proof. For r ≥ 2, we have (with the convention that π0 = π1 ≡ 1):

πr ≤ πbr/2cπdr/2e +

br/2c∑i=1

dr/2e∑j=1

1

4i+j−1πbr/2c−iπdr/2e−j, (38)

where we first sum πr(I) over I ⊆ {1, . . . , r − 1} such that br/2c ∈ I, and then sum πr(I)

over I ⊆ {1, . . . , r−1} such that br/2c 6∈ I and the closest elements in I∪{0, r} to br/2c are

br/2c−i and br/2c+j; we use the fact that for I ′ = {i1, i2, . . . , il} and I ′′ = {il+1, il+2, . . . , iL}with 0 < i1 < i2 < · · · < il ≤ il+1 < il+2 < · · · < iL < r, we have

πr(I′ ∪ I ′′) ≤

πil(I ′ − {il}) · πr−il+1(I ′′ − {il+1}) il = il+1

πil(I′ − {il}) · πr−il+1

(I ′′ − {il+1})/4il+1−il−1 il < il+1

. (39)

27

Therefore, for any l ≤ br/2c we have

πr ≤ πbr/2cπdr/2e +l∑

i=1

l∑j=1

1

4i+j−1πbr/2c−iπdr/2e−j +

l∑i=1

∞∑j=l+1

1

4i+j−1+

∞∑i=l+1

∞∑j=1

1

4i+j−1

≤ 13

9max−1≤j≤l

π2br/2c−j +

8

9

(1

4

)l, (40)

where we use Lemma 2 which implies that πbr/2c−i and πdr/2e−j are always less than or equal

to 1.

We can directly calculate πr when r is small. We use the following facts from our

calculations: π7 ≈ 0.595 and πr < 0.595 for 8 ≤ r ≤ 17.

Define the function

Hl(y) =13

9y2 +

8

9

(1

4

)l, (41)

and let (Hl)i be Hl iterated i times.

First let l = 2. When y is in between 0.0609154 and 0.631392 which are the two solutions

to Hl(y) = y, we have Hl(y) < y and limi→∞(Hl)i(y) = 0.0609154. By Equation (40) and

our calculations of πr for 7 ≤ r ≤ 17, we have πr < Hl(0.595) for r ≥ (7 + 2) × 2 = 18;

πr < (Hl)2(0.595) for r ≥ (18 + 2) × 2 = 40; πr < (Hl)

3(0.595) for r ≥ (40 + 2) × 2 = 84;

and so on. Therefore, we have lim supr→∞ πr ≤ limi→∞(Hl)i(0.595) = 0.0609154.

We then assume progressively larger values of l and use similar reasoning in the above

paragraph to conclude that lim supr→∞ πr ≤ 0.

Proposition 1 implies that the probability of sequential stability vanishes with r in the

exponential model. Section A.3 leverages this fact to prove that the probability of sequential

stability vanishes with n in our actual model.

5 Flexible Transfers

In this section we supplement the fixed-transfer model with transfers that are endogenously

determined as a part of the stable matching. We show that flexible transfers eliminate the

incentives to match early and restore sequential stability.

Theorem 3. Consider any n ≥ 1 and k ≥ 1 in the flexible-transfer regime. For any

selection of stable matching (µ2, P 2) in the second period, the matching scheme consisting of

no early matching (µ = ∅, µ2, P 2) is sequentially stable for every realization of first-period

28

types. Moreover, when (µ2, P 2) is always the woman-optimal stable matching, for almost

every realization of first-period types (µ, µ2, P 2) is sequentially stable if and only if µ = ∅.17

Proof. Fix a realization of types in the first period, consider no early matching: µ = ∅.For any man mi and woman wj in the first period, and for any type realization of the new

arrivals, suppose that mi is ranked r-th and wj is ranked s-th in the second period (among

the new arrivals and the first-period agents), i.e., mi = m2r and wj = w2

s . By the assumption

that a stable matching (µ2, P 2), where µ2 is assortative, is implemented in the second period,

we have the no-blocking condition:

(U(w2

r | m2r) + P 2(m2

r, w2r))

+(V (m2

s | w2s)− P 2(m2

s, w2s))≥ U(w2

s | m2r) + V (m2

r | w2s)

= U(wj | mi) + V (mi | wj) (42)

Take expectation over the type realization of the new arrivals in (42) gives (where L here

is the list of all first-period agents since µ = ∅):

U(L,mi;P2) + V(L,wj;P

2) ≥ U(wj | mi) + V (mi | wj), (43)

i.e., man mi and woman wj do not have a joint incentive to match early. This proves the

first part of the theorem.

Now suppose (µ2, P 2) is always the woman-optimal stable matching. Let L ⊇ {mi, wj}now be a list of first-period agents who wait for period 2. If the first-period types are all

distinct (happens with probability 1), then inequality (42) is strict unless r = s+ 1 or r = s

(where mi is ranked r-th and wj is ranked s-th in the second period), which fails to occur

with a positive probability over the new arrivals. Thus, inequality (43) is also strict. Thus,

no pair of first-period agents can have a joint incentive to match early (inequality (19) in

Definition 4 cannot hold).

Theorem 3 presents a sharp contrast to the results from the fixed-transfer regime: with

flexible transfers in period 2, none of the agents have an incentive to deviate from the second-

period stable matching (in contrast to Theorem 1), which implies that everyone participat-

ing in the second-period stable matching is sequentially stable (in contrast to Theorem 2).

Moreover, under the woman-optimal implementation of the second-period transfers everyone

participating in period 2 is the unique sequentially stable matching scheme; we conjecture

17In the second part of Theorem 3 we select (µ2, P 2) to be the woman-optimal stable matching for everyrealization of types in period 2 (those who wait from period 1 plus the new arrivals). Clearly, the same resultholds if (µ2, P 2) is always the man-optimal stable matching.

29

that the same result also holds if the second-period matching is always the median stable

matching. Intuitively, the flexible transfers mitigate the risk in period 2: when a high type

woman is assortatively matched to a low type man in period 2 (because of strong women and

weak men in the new arrivals), the stable matching gives the high type woman a sufficiently

large fraction of her joint surplus with the low type man, such that her fraction of surplus

dominates all incentives to match early.

Theorem 3 implies that the timing problems identified in Section 4 (the prevalence of

early matching incentives and the failure of sequential stability) are caused by the lack of

flexible transfers. Theorem 3 thus gives a rationale for the desirability of flexible transfers

in matching.

Example 4. We now show that early matching can occur in a sequentially stable matching

scheme when the stable matching is not “consistently” implemented in period 2. This ex-

ample clarifies the role of man/woman-optimal stable matching in Theorem 3. Consider the

setting of Example 3 (Section 3.3) with n = 1 and k = 1. Let (m,w) denote the types of

agents in the first period and (m2, w2) the new arrivals’ types. Instead of always having a

woman-optimal stable matching in period 2 as in Example 3, in the situation when man m

and woman w wait for period 2, suppose that the man-optimal stable matching is imple-

mented in period 2 when woman w is doing better than man m (w > w2 and m2 > m), and

that the woman-optimal stable matching is implemented in period 2 when man m is doing

better than woman w (m > m2 and w2 > w). In all other cases, assume an arbitrary stable

matching is implemented in period 2 (It does not matter because in these cases man m is

assortatively matched to woman w, if they wait for period 2.)

Then, for every realization of the first-period type (m,w), the early matching of µ =

{(m,w)} is sequentially stable given the second-period stable matchings described in the

previous paragraph: by those specific choices of stable matching in period 2, the total surplus

of m · w from matching early is exactly equal to the sum of utilities of man m and woman

w in period 2 for every realization of (m2, w2); for example, in the woman-optimal stable

matching which is implemented when w < w2 and m2 < m, man m is exactly indifferent

between matching with w2 and deviating to match with w (see the specification of transfers in

Example 3). Of course, by the first part of Theorem 3 no early matching is also sequentially

stable in this example.

30

5.1 An intermediate regime

In this subsection we analyze an intermediate regime in which flexible transfers are possible

in period 1, but not in period 2. The motivation is that while some institutional constraint

may impose a rigid schedule of transfers in the formal matching institution (period 2), if

agents match outside of the formal institution they may be free to negotiate any division of

the surplus (period 1). This intermediate regime is plausible if we interpret early matching as

attempts to cheat or to circumvent the formal matching institution of period 2: if agents are

cheating then it is likely that they also ignore the institutionally fixed schedule of transfers.

We show that flexible transfers in period 1 alone is not sufficient to prevent early matching

nor to guarantee sequential stability. These results reinforce the importance of the flexible

transfers in the formal matching institution (i.e., period 2 in our model). Moreover, flexible

transfers in period 1 alone may enable a kind of cross match of higher-ranked woman and

lower-ranked man (or vice versa) in early matching that would not be possible with fixed

transfers.

In the second period of this intermediate regime, an assortative matching with fixed

transfers is implemented — when man m2i is assortatively matched to woman w2

i in period

2, man m2i gets U(w2

i | m2i ) and woman w2

i gets V (m2i | w2

i ). Then, given a list L of first-

period agents waiting to period 2, man mi ∈ L has an expected utility of U(L,mi) (defined

in (9)) from the assortative matching in period 2, and woman wi ∈ L has an expected utility

of V(L,wi) (defined in (10)). In the first period, a pair of agents can negotiate any division

of surplus if they choose to match early. Then, man mi and woman wi have a strict incentive

to match early if their joint surplus (the amount available to divide) is greater than the sum

of their expected utilities from period 2:

U(wi | mi) + V (mi | wi) > U(L,mi) + V(L,wi). (44)

As before, we first consider the benchmark where all first-period agents wait for the

assortative matching in the second period, i.e., let L be the list of all first-period agents in

(44). Let ξn be the number of first-period pairs (mi, wi) with a strict incentive to match

early given this benchmark (i.e., inequality (44) holds). We have the following analogue of

Theorem 1:

Theorem 4. Fix k ≥ 1 pairs of new arrivals. Then in the intermediate regime, the expected

fraction of first-period pairs with an incentive to match early, E[ξn]/n, tends to 1/2 as the

number n of first-period agents tends to infinity.

31

Proof. We omit the proof because it is virtually identical to that of Theorem 1.

Theorem 4 reveals that flexible transfers in period 1 actually exacerbates the prevalence

of incentives to match early: the early matching fraction increases from 1/4 with fixed

transfers (Theorem 1) to 1/2 in the this intermediate regime. Intuitively, with flexible

transfers in period 1 we no longer need both agents to prefer the early matching (as we do

with fixed transfers): one side who is eager to match early now has the option of contributing

some transfers to the other side (who might be unwilling otherwise) to “sweeten” the early

matching deal. Since we now have one early-matching incentive condition instead of two, we

have the early-matching probability of 1/2 instead of 1/4.

Next, we adapt the definition of sequential stability, Definition 2, for this intermediate

regime. Since agents anticipate an assortative matching with fixed transfers in period 2,

condition (1) in Definition 2 is unchanged. Condition (2) in Definition 2, the incentive for

not matching early, is now:

U(L(µ),mi) + V(L(µ), wj) ≥ U(wj | mi) + V (mi | wj), (45)

since agents can negotiate transfers in period 1. Similarly, condition (3) in Definition 2, the

incentive for matching early, becomes:

U(wj | mi) + V (mi | wj) ≥ U(L(µ) ∪ {mi, wj},mi) + V(L(µ) ∪ {mi, wj}, wj). (46)

Given this definition sequential stability in the intermediate regime, one can show that

the realization of types in Example 2 (Section 3.2) still does not admit a sequentially stable

matching scheme. Indeed, because of the symmetry of man and woman in Example 2, we

have U(wi | mi) = V (mi | wi) as well as U(L(µ),mi) = V(L(µ), wi) for any assortative early

matching µ, and hence U(L(µ),mi) + V(L(µ), wi) ≥ U(wi | mi) + V (mi | wi) if and only

if U(L(µ),mi) ≥ U(wi | mi). Therefore, the sequential instability of any assortative early

matching follows directly from the argument when the transfers are fixed (see Section 3.2).

It is also easy to check that any non-assortative early matching is not sequentially stable (for

example, the cross match in Example 5 cannot work here). Thus, flexible transfers in period

1 alone cannot guarantee sequential stability. We conjecture that Theorem 2 continues to

hold with flexible transfers in period 1.

Finally, we illustrate that flexible transfers in period 1 may enable a kind of cross match

between a high-ranked woman and a low-ranked man in early matching that would not be

possible with fixed transfers. This example is in the same spirit as Theorem 4 and shows

32

that flexible transfers in period 1 expand the possibilities of early matching.

Example 5. Suppose that n = 2, k = 1, F = G = U[0, 1] and U(w | m) = V (m | w) = mw/2.

Fix the realization of

m2 =9

10,m1 =

4

10, w2 =

4.2

10, w1 =

1

10, (47)

in the first period. We claim that the early matching µ = {(m1, w2)} is sequentially stable

with flexible transfers in period 1. Notice that the early matching µ = {(m1, w2)} cannot

be sequentially stable with fixed transfers, because of the presence of m2 and w1 in period

2 and that k = 1, w2 will always be matched to a type higher than m1 in period 2 (and

potentially even higher than m2 = 9/10), so with fixed transfers woman w2 will never have

an incentive to match early with man m1. On the other hand, man m1 is always matched to

someone worse than w2 in period 2 (and potentially even worse than w1 = 1/10), so woman

w2 could leverage her advantage and m1’s disadvantage in period 2 to match early with m1

and demand a large share of the surplus from this early matching. Indeed with transfers in

period 1 this is sequentially stable for the realization of types in (47), and both man m1 and

woman w2 strictly benefits from this early matching. (Detail can be found in Appendix B.)

One can show that for the realization in (47), µ = {(m1, w2)} is the unique sequentially

stable early matching with flexible transfers in period 1, and µ = ∅ is the unique sequentially

stable early matching with fixed transfers. Therefore, fixed transfers may lead to greater

efficiency in equilibrium than flexible transfers in period 1 alone.

6 Discussion

6.1 Tradeoffs in Large Markets

When the types of agents are distributed on a bounded interval and given a fixed k, the

arrangement in which nobody matches early (µ = ∅) becomes approximately stable in the

fixed-transfer regime as n tends to infinity, because k pairs of arrivals could displace the ex-

isting agents by at most k ranks, and the difference between the types of two consecutively

ranked agents tends to 0 at a rate of Op(1/n) given a bounded type distribution. Theo-

rem 1 and Theorem 2 are of interest for bounded type distribution because they identify

novel tradeoffs, as n gets large, between the decrease in the impact of the new arrivals on

the existing agents on the one hand, and the increase in the number of agents wanting to

matching early and the increase in the probability of sequential instability caused by the

new arrivals on the other hand. The main message of our paper is that such tradeoff does

33

not exist when the transfers are flexible.

To understand which of these effects dominates we must study the rates at which these

effects appear or vanish as n becomes large. Theorem 1 implies that the expected number

of pairs with an incentive to match early increases at least linearly with n. More research is

needed to quantify the rate at which the probability of sequential instability tends to 1 with

n. The Monte Carlo results presented in Figure 3 suggest that the probability of sequential

instability increases to 1 linearly with n when the type distribution is uniform and the values

are multiplicative.

Theorem 1 and Theorem 2 are also true for unbounded type distributions. When the

agents’ types have unbounded distribution and when the surplus function features large

complementarities between the high-type agents (e.g., U(w | m) = V (m | w) = mw/2),

nobody matching early is not approximately stable under k = 1 as n tends to infinity,

since even a small increase in the type of the matched partner can generate a large increase

in surplus for the high-type agent due to their complementarities. Unbounded types are

plausible when the number of agents tends to infinity, since it allows the possibility of

exceptional talent/productivity in large population.

Finally, we note that under k = 1 and a bounded distribution of types, the decision by a

first-period agent to wait for period 2 is not approximately a dominant strategy even when

n is large, because waiting for period 2 could be very costly for an agent when others below

him have all committed to early matchings, thus exposing this agent to a large risk of being

matched to a low type in period 2.

6.2 Interpreting Theorem 2

As we have noted the failure of sequential stability in the fixed transfers regime has a “match-

ing pennies” logic. To sustain an equilibrium in a matching pennies game convexity in the

form of mixed strategy is needed. In our setting a natural source of convexity is the trans-

fers between the matched partners, and indeed we show that flexible transfers can sustain

sequential stability. So we view Theorem 2 as an argument for introducing flexible transfers

to stabilize the market (and to prevent early matching). In practice, it is quite plausible

that an employer can promise an employee who proposes early matching to pay the employee

more if they are matched in period 2 and if the employee is in high demand then. This kind

of promise would introduce flexible transfers in period 2 and would help stabilize the market.

Comparing with transfers, explicit randomization in early matching decisions is less nat-

ural in our setting. On the other hand, randomization in early matching decisions could be

34

viewed as due to uncertainty about an agent’s type in period 1, in the spirit of Harsanyi’s

purification theorem. Since in this paper we focus on the role of transfers in early matching,

we leave exploring the purification argument to future research.

6.3 Assumption on n and k

For our fixed-transfer results we have assumed a large n in period 1 and a small k in period

2. This assumption corresponds to the natural benchmark in which most of the agents (i.e.,

those in period 1) endogenously choose their time of matching; the small k arrivals in period

2 can be interpreted as a small shock to the market. The case of large n and large k is

also interesting and raises intriguing theoretical questions (see Equation (70) in Appendix C

for the expected utility with arbitrary n and k). The case of large n and large k could

be approximated with continuums of agents in both periods, which is another interesting

direction for the future.

35

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38

Appendix

A Proofs

A.1 Preliminary Lemmas

We use the following lemmas in the proofs of Theorem 1 and Theorem 2.

Lemma 3 (Chernoff).

P(∣∣∣∣F (mi)−

i

n

∣∣∣∣ ≥ ε

)≤ 2 exp(−2ε2n)

for any ε > 0.

Proof of Lemma 3. Let u1 ≤ . . . ≤ un be the order statistics of n i.i.d. U[0, 1] random

variables, and set mi ≡ F−1(ui) where F−1(u) ≡ inf{x ∈ R : F (x) ≥ u}. Since the density

f is everywhere positive, we have F (mi) = ui.

Clearly,

P(|ui − i/n| ≥ ε) ≤ P(ui ≥ i/n+ ε) + P(ui ≤ i/n− ε).

By definition, we have

P(ui ≤ i/n− ε) = P

(n∑j=1

1(zj ≤ i/n− ε) ≥ i

)

where z1, . . . , zn are n i.i.d. U[0, 1] random variables.

We now apply a standard Chernoff bound to i.i.d. random variables 1(zj ≤ i/n − ε)’s

(e.g., Alon and Spencer (2008), Theorem A.1.4):

P

(n∑j=1

1(zj ≤ i/n− ε) ≥ i

)= P

(n∑j=1

(1(zj ≤ i/n− ε)− (i/n− ε)) ≥ nε

)≤ exp(−2ε2n).

39

Similarly,

P (ui ≥ i/n+ ε) ≤ P

(n∑j=1

1(zj ≥ i/n+ ε) ≥ n− i

)

= P

(n∑j=1

(1(zj ≤ i/n+ ε)− (1− i/n− ε)) ≥ nε

)≤ exp(−2ε2n).

Lemma 4 connects the differences between consecutive U[0, 1] order statistics to expo-

nential random variables. It is well-known, and the proof can be found in Pyke (1965).

Lemma 4. Let u1 ≤ . . . ≤ un be the order statistics of n i.i.d. U[0, 1] random vari-

ables. Then, (1 − un, un − un−1, un−1 − un−2, . . . , u2 − u1, u1) has the same distribution

as (x1/x, x2/x, . . . , xn+1/x), where x1, . . . , xn+1 are i.i.d. exponential random variables and

x ≡∑n+1

i=1 xi.

The following lemma is an easy exercise in integration:

Lemma 5. Suppose that x1, . . . xl, y are i.i.d. unit exponential random variables. Then for

any c > 0,

P

(c

l∑i=1

xi ≤ y

)= (1 + c)−l.

Proof of Lemma 2 (page 27). Fix a realization (mi, wi)0≤i≤r such that

mj −mi 6= ml −mj, wj − wi 6= wl − wj

for all 0 ≤ i < j < l ≤ r. We will prove that (mi, wi)0≤i≤r admits at most one assortative

early matching that is sequentially stable in the sense of Definition 5.

Let I = {i1, . . . , iL} and I ′ = {i′1, . . . , i′K}, where i0 ≡ 0 < i1 < · · · < iL < r ≡ iL+1 and

i′0 ≡ 0 < i′1 < · · · < i′K < r ≡ i′K+1. Suppose that I 6= I ′. For the sake of contradiction,

suppose that both I and I ′ are sequentially stable for (mi, wi)0≤i≤r in the sense of Definition 5.

First, find the smallest l̄ ≤ min(L+ 1, K + 1) in which il̄ 6= i′l̄. Without loss assume that

il̄ < i′l̄. This means that il̄ ∈ I but il̄ 6∈ I ′. Since both I and I ′ are sequentially stable, for

40

the couple of index il̄, we have simultaneously:

mil̄−mil̄−1

< mil̄+1−mil̄

or wil̄ − wil̄−1< wil̄+1

− wil̄

and (since il̄−1 = i′l̄−1

< il̄ < i′l̄)

mil̄−mi′

l̄−1> mi′

l̄−mil̄

and wil̄ − wi′l̄−1> wi′

l̄− wil̄

which imply that il̄+1 > i′l̄.

Now consider index i′l̄: analogues of the above two inequalities imply that i′

l̄+1> il̄+1.

That i′l̄+1

> il̄+1 in turn implies that il̄+2 > i′l̄+1

(considering index il̄+1), and so on. Even-

tually we would conclude either i′l̄+j

> iL+1 = r or il̄+j > i′K+1 = r, which are both

impossible.

A.2 Proof of Theorem 1

Let qi,n denote the probability that the first-period pair (mi, wi) has an incentive to match

early, given that everyone is waiting for the assortative matching in the second period.

Let Pn denote the probability measure over the types in the first period (and En for the

corresponding expectation). Finally, let bxc be the largest integer less than or equal to x.

Fix an arbitrary δ > 0. We prove that there exists N such that for any n ≥ N we have:∣∣∣∣qi,n − 1

4

∣∣∣∣ ≤ δ, for every δn < i < (1− δ)n. (48)

As a consequence, for n ≥ N we have:

En[ξn] =n∑i=1

qi,n ≥bn(1−δ)c∑i=bnδc

qi,n ≥(

1

4− δ)

(1− 2δ)n,

which implies a lower bound of 1/4 for pairs with an incentive to match early. The derivation

of the upper bound is analogous and proves the theorem.

We now prove (48). By waiting to the second period, assuming everyone else does so as

41

well, man mbnpc’s expected utility is:

∑0≤i<j≤k

(k

i

)F (mbnpc)

i(1− F (mbnpc))k−i(k

j

)G(wbnpc)

j(1−G(wbnpc))k−j

× E(G−(wbnpc))⊗j [U(w2bnpc−(j−i) | mbnpc)]

+∑

0≤j<i≤k

(k

i

)F (mbnpc)

i(1− F (mbnpc))k−i(k

j

)G(wbnpc)

j(1−G(wbnpc))k−j

× E(G+(wbnpc))⊗(k−j) [U(w2

bnpc+(i−j) | mbnpc)]

+∑

0≤i≤k

(k

i

)F (mbnpc)

i(1− F (mbnpc))k−i(k

i

)G(wbnpc)

i(1−G(wbnpc))k−i

× U(wbnpc | mbnpc),

where (G−(wbnpc))⊗j is the probability measure of j i.i.d. random variables from the con-

ditional distribution G(x)/G(wbnpc), x ≤ wbnpc; w2bnpc−(j−i) is the (bnpc − (j − i))-th lowest

woman among the j new women (from the distribution G(x)/G(wbnpc)) and the first period

women w1 ≤ . . . ≤ wn; and likewise for other terms.

Comparing this to U(wbnpc | mbnpc), we see that man mbnpc strictly prefers to match early

with woman wbnpc if and only if:

∑0≤i<j≤k

(k

i

)F (mbnpc)

i(1− F (mbnpc))k−i(k

j

)G(wbnpc)

j(1−G(wbnpc))k−j

×(U(wbnpc | mbnpc)− E(G−(wbnpc))⊗j [U(w2

bnpc−(j−i) | mbnpc)])

>∑

0≤j<i≤k

(k

i

)F (mbnpc)

i(1− F (mbnpc))k−i(k

j

)G(wbnpc)

j(1−G(wbnpc))k−j (49)

×(E(G+(wbnpc))

⊗(k−j) [U(w2bnpc+(i−j) | mbnpc)]− U(wbnpc | mbnpc)

),

which has the interpretation of comparing man mbnpc’s downside in period 2 with his upside.

We have the following uniform-convergence result, and analogously for the terms on the

RHS of (49) and in the inequality for woman wbnpc’s early-matching incentive.

42

Lemma 6. Fix j > i. Without loss of generality, let v1 ≤ . . . ≤ vn be the order statistics of

n i.i.d. U[0, 1] random variables, and set wi′ ≡ G−1(vi′), 1 ≤ i′ ≤ n.

For every ε > 0, we have

supp∈[δ,1−δ]

Pn

∣∣∣∣∣∣∣∣∣∣∣

F (mbnpc)i(1− F (mbnpc))

k−iG(wbnpc)j(1−G(wbnpc))

k−j

×(U(wbnpc | mbnpc)− E(G−(wbnpc))⊗j [U(w2

bnpc−(j−i) | mbnpc)])

pi(1− p)k−ipj(1− p)k−jU ′(G−1(p) | F−1(p))(vbnpc − vbnpc−(j−i))/g(G−1(p))− 1

∣∣∣∣∣∣∣∣∣∣∣> ε

converges to zero with n.

Proof. Lemma 6 is a consequence of Lemma 3 and of the uniform continuities of f(x), g(y),

U(y | x) and V (x | y) for (x, y) ∈ [F−1(δ/2), F−1(1− δ/2)]× [G−1(δ/2), G−1(1− δ/2)].

Let

(a−k, . . . , a−1, a1, . . . , ak, α−k, . . . , α−1, α1, . . . , αk)

be 4k i.i.d. unit-exponential random variables. Define

ν(p, C) ≡ P

C∑

0≤i<j≤k(ki

)pi(1− p)k−i

(kj

)pj(1− p)k−j ·

∑−1l=i−j al

>∑

0≤j<i≤k(ki

)pi(1− p)k−i

(kj

)pj(1− p)k−j ·

∑i−jl=1 al,

C∑

0≤i<j≤k(ki

)pi(1− p)k−i

(kj

)pj(1− p)k−j ·

∑−1l=i−j αl

>∑

0≤j<i≤k(ki

)pi(1− p)k−i

(kj

)pj(1− p)k−j ·

∑i−jl=1 αl

.

By Lemma 6 and Lemma 4, for any ε > 0, there exists N such that

ν

(p,

1− ε1 + ε

)− ε ≤ qbnpc,n ≤ ν

(p,

1 + ε

1− ε

)+ ε

for all p ∈ [δ, 1− δ] and n ≥ N .

Furthermore, ν(p, C) converges to ν(p, 1) = 1/4 as C → 1, uniformly in p ∈ [δ, 1 − δ],which implies our conclusion.

A.3 Proof of Theorem 2

Throughout the proof we abuse our terminology and say that an early matching is sequen-

tially stable if it, together with the assortative matching in the second period, forms a

sequentially stable matching scheme in the sense of Definition 2; moreover, we focus on in-

43

dices instead of type realizations and let µ ⊆ {1, . . . , n}2. As in the proof of Theorem 1, we

use Pn to denote the probability measure over the types in the first period (and En for the

corresponding expectation), and use bxc to denote the largest integer less than or equal to

x. And let

M≡

{µ ⊂ {1, . . . , n}2 :

(i, j), (i, j′) ∈ µ⇒ j = j′,

(i′, j), (i, j) ∈ µ⇒ i = i′

}be the set of all possible early matchings.

For any µ ∈M, let

Cµ ⊆ {(m1, . . . ,mn, w1, . . . , wn) ∈ R2n | mn ≥ . . . ≥ m1, wn ≥ . . . ≥ w1}

be the set of first-period types that admit µ as a sequentially stable early matching. Theo-

rem 2 states that under the assumption of k = 1, we have

limn→∞

Pn

( ⋃µ∈M

Cµ

)= 0.

We first note that under the assumption k = 1, any sequentially stable early matching µ

satisfies the following “intermediate” property:

(i, j) ∈ µ, i ≤ l ≤ j or j ≤ l ≤ i

=⇒ there exists i′ and j′ such that (i′, l) ∈ µ and (l, j′) ∈ µ (Int)

This property follows from the fact that if µ is sequentially stable and (i, j) ∈ µ, then

if one of them breaks the early matching, man i and woman j must be of the same rank

among those who wait (according to µ) plus themselves. Suppose otherwise, say man i is of

a lower rank than woman j, then woman j cannot have incentive to match early with man i

because the new arrivals can change woman j’s ranking by at most one place in the second

period (recall that k = 1), so in any case she gets a better (or equal) match than mi.

Define

Mint ≡ {µ ∈M : µ satisfies property (Int)}.

We have that Cµ = ∅ for any µ 6∈ Mint.

Notice that if µ ∈ Mint, then the set of man-ranks who do not match early under µ

44

equals the set of woman-ranks who do not match early under µ:

I(µ) ≡ {i ∈ {1, . . . , n} : ∀j, (i, j) 6∈ µ} = {j ∈ {1, . . . , n} : ∀i, (i, j) 6∈ µ}. (50)

We fix an arbitrary p ∈ (0, 1) throughout the proof of Theorem 2.

For fixed integers s > 0 and t > 0, define:

I ≡ {I : I ⊆ {1, . . . , n}}.

Ij,l ≡ {I ∈ I : bnpc+ t+ j ∈ I and bnpc − t− l ∈ I}. (51)

I ′ ≡

{I ∈ I :

min({i ∈ I : i ≥ bnpc+ t}) > bnpc+ t+ s, or

max({i ∈ I : i ≤ bnpc − t}) < bnpc − t− s

}.

In words, I ′ is the set of agents’ ranks in which there is a “gap” of size at least s, either

starting at bnpc+ t or ending at bnpc − t.Clearly,

I = I ′ ∪⋃

0≤j≤s0≤l≤s

Ij,l.

We can further divide I ′ into I ′1 and I ′2: I = I ′1 ∪ I ′2, where

I ′1 ≡ {I ∈ I : min({i ∈ I : i ≥ bnpc+ t}) > bnpc+ t+ s} , (52)

I ′2 ≡ {I ∈ I : max({i ∈ I : i ≤ bnpc − t}) < bnpc − t− s} .

Let

CI ≡⋃

µ∈Mint,I(µ)=I

Cµ

for I ∈ I, and

C(I ′′) ≡⋃I∈I′′

CI

for I ′′ ⊆ I.Clearly, we have

C(I) =⋃µ∈M

Cµ.

45

On the other hand,

Pn(C(I)) ≤ Pn(C(I ′1)) + Pn(C(I ′2)) +∑

0≤j≤s0≤l≤s

C(Ij,l).

Therefore,

lim supn→∞

Pn(C(I)) ≤ lim supn→∞

Pn(C(I ′1)) + lim supn→∞

Pn(C(I ′2))

+∑

0≤j≤s0≤l≤s

lim supn→∞

Pn(C(Ij,l)). (53)

In the next two subsections we show that for a fixed s, lim supn→∞ Pn(C(Ij,l)) goes to 0

as t goes to infinity for any j and l, at a rate independent of s (Section A.3.1, Equation (58)

and Proposition 1); and that for a fixed t, lim supn→∞ Pn(C(I ′1)) and lim supn→∞ Pn(C(I ′2))

go to 0 as s goes to infinity, at a rate independent of t (Section A.3.2, Equations (65) and

(66)). This implies that by choosing s and t sufficiently large, we can make the left hand

side of (53), which is independent of s and t, as close to zero as we want. Thus, the left hand

side of (53) must be exactly zero, which proves Theorem 2.

A.3.1 Reduction to the Exponential Model of Section 4.2.2

We first bound the term lim supn→∞ Pn(C(Ij,l)) in (53). An early matching µ with I(µ) ∈ Ij,lhas the property that men and women of ranks bnpc+t+j and bnpc−t−l do not match early

under µ. Thus, types in C(Ij,l) satisfy the property that assuming that men and women of

ranks bnpc+ t+ j and bnpc− t− l wait to the second period, men and women between ranks

bnpc+ t+ j and bnpc − t− l admit a sequentially stable early matching among themselves.

Therefore, in this subsection we solve the following local problem. Fix an integer r >

1. Assuming that exogenously men and women of ranks bnpc and bnpc + r wait to the

second period, what is the probability that agents between ranks bnpc and bnpc + r admit

a sequentially stable early matching among themselves? 18

Denote

U ′(w | m) ≡ ∂U(w | m)

∂w, V ′(m | w) ≡ ∂V (m | w)

∂m. (54)

We first note that given k = 1, the expected utility of mi in the second period is (where

18It will be obvious that we get the same result by assuming that men and women of ranks bnpc+ i andbnpc+ i+ r wait to the second period, for any fixed i.

46

w+ > wi > w− are the types of women who are going to the second period (women in the

list L) and who are just around the rank of mi):

U(L,mi) =

(F (mi)G(wi) + (1− F (mi))(1−G(wi))

)U(wi | mi) (55)

+ (1− F (mi))

(U(w− | mi)G(w−) +

∫ wi

w−

U(x | mi)g(x) dx

)+ F (mi)

(U(w+ | mi)(1−G(w+)) +

∫ w+

wi

U(x | mi)g(x) dx

),

where the first term represents the events in which mi is matched to wi in the second period,

the second term represents events in which mi is matched to a worse type than wi, and the

last term represents the events in which mi is matched to a better type than wi. Clearly, a

symmetric formula holds for woman wi.

It is easy to use integration by parts to verify that mi has strict incentive to match early

with wi (i.e., U(wi | mi) dominates (55)) if and only if:

(1− F (mi))

∫ wi

w−

U ′(x | mi)G(x) dx > F (mi)

∫ w+

wi

U ′(x | mi)(1−G(x)) dx, (56)

which has the interpretation of comparing mi’s downside in period 2 with his upside.

Before putting (56) to use, let us adapt some of the previous notations to the present,

local setting. Let

M(r) ≡

{µ ⊂ {1, . . . , r − 1}2 :

(i, j), (i, j′) ∈ µ⇒ j = j′,

(i′, j), (i, j) ∈ µ⇒ i = i′

};

an early matching among agents between ranks bnpc and bnpc + r can be represented a

µ ∈ M(r): man of rank bnpc + i matches early with woman of rank bnpc + j according to

µ if and only if (i, j) ∈ µ.

And as before, define

M(r)int ≡ {µ ∈M(r) : µ satisfies property (Int)},

where the property (Int) is defined in page 44.

Finally, for each µ ∈M(r)int, define

Ir(µ) ≡ {i ∈ {1, . . . , r − 1} : ∀j, (i, j) 6∈ µ} = {j ∈ {1, . . . , r − 1} : ∀i, (i, j) 6∈ µ}.

47

to be the set of ranks of men/women who wait to second period according to µ.

Given (56), we can simplify Definition 2:

Definition 6. Fix an ordered list of types P =((mi)bnpc≤i≤bnpc+r, (wi)bnpc≤i≤bnpc+r

)and an

early matching µ ∈ M(r)int. Suppose that Ir(µ) = {il : 1 ≤ l ≤ L}, where 0 ≡ i0 < i1 <

. . . < iL < iL+1 ≡ r. Then, early matching µ is sequentially stable for P if:

1. for every 1 ≤ l ≤ L, we have either

(1− F (mbnpc+il))

∫ wbnpc+il

wbnpc+il−1

U ′(x | mbnpc+il)G(x) dx

≤F (mbnpc+il)

∫ wbnpc+il+1

wbnpc+il

U ′(x | mbnpc+il)(1−G(x)) dx,

or

(1−G(wbnpc+il))

∫ mbnpc+il

mbnpc+il−1

V ′(x | wbnpc+il)F (x) dx

≤G(wbnpc+il)

∫ mbnpc+il+1

mbnpc+il

V ′(x | wbnpc+il)(1− F (x)) dx,

2. for the couple (i, j) ∈ µ who matches early (there exists a unique l such that il < i <

il+1 and il < j < il+1), we have

(1− F (mbnpc+i))

∫ wbnpc+j

wbnpc+il

U ′(x | mbnpc+i)G(x) dx

≥F (mbnpc+i)

∫ wbnpc+il+1

wbnpc+j

U ′(x | mbnpc+i)(1−G(x)) dx,

and

(1−G(wbnpc+j))

∫ mbnpc+i

mbnpc+il

V ′(x | wbnpc+j)F (x) dx

≥G(wbnpc+j)

∫ mbnpc+il+1

mbnpc+i

V ′(x | wbnpc+j)(1− F (x)) dx.

For µ ∈M(r)int, let

Dµ ⊆

{((mi)bnpc≤i≤bnpc+r, (wi)bnpc≤i≤bnpc+r

)∈ R2(r+1) :

mbnpc+r ≥ . . . ≥ mbnpc,

wbnpc+r ≥ . . . ≥ wbnpc

}

48

be the set of types in between ranks bnpc and bnpc+ r that admit µ as a sequentially stable

early matching according to Definition 6.

For each I ⊆ {1, . . . , r − 1}, let GI ⊆ R2(r+1) be the types that make I sequentially

stable according to Definition 5 in Section 4.2.2. Let P̂r be the probability measure of the

exponential model described in Section 4.2.2 and recall the definition of πr in Equation (36).

Proposition 2. Fix p ∈ (0, 1) and integer r ≥ 2. We have

limn→∞

Pn

⋃µ∈M(r)int

Dµ

= P̂r

⋃I⊆{1,...,r−1}

GI

= πr. (57)

Proof. The proof is omitted because it is identical to that of Theorem 1. Essentially, because

mbnpc+r −mbnpc and wbnpc+r −wbnpc converge to zero in probability with n, the condition in

Point 1 of Definition 6 becomes approximately

wbnpc+il − wbnpc+il−1≤ wbnpc+il+1

− wbnpc+il or mbnpc+il −mbnpc+il−1≤ mbnpc+il+1

−mbnpc+il ,

and the condition in point 2 of Definition 6 becomes approximately

wbnpc+j − wbnpc+il ≥ wbnpc+il+1− wbnpc+j and mbnpc+i −mbnpc+il ≥ mbnpc+il+1

−mbnpc+i,

when n is large. (Compare with Definition 5 for the definition of GI in the exponential

model.)

Now going back to bounding (53): by Proposition 2 we have

lim supn→∞

Pn(C(Ij,l)) = π2t+j+l (58)

because positions bnpc+t+j and bnpc−t−l are of distance 2t+j+l apart (cf. Equation (51)).

Therefore, Proposition 1 proves that lim supn→∞ Pn(C(Ij,l)) goes to 0 as t goes to infinity,

for any fixed s > 0, 0 ≤ j ≤ s and 0 ≤ l ≤ s.

A.3.2 Consecutive pairs matching early

In this subsection we give the required bounds for lim supn→∞ Pn(C(I ′1)) and lim supn→∞ Pn(C(I ′2))

in (53). Recall that p is an arbitrary percentile in (0, 1). Intuitively, we are removing the “lo-

cal” assumption of the previous subsection that men and women of ranks bnpc and bnpc+ r

49

exogenously wait to the second period.

Let us focus lim supn→∞ Pn(C(I ′1)); the bound for lim supn→∞ Pn(C(I ′2)) is analogous.

Let t̄ ≡ t+ s+ 1, m0 ≡ m > −∞ and w0 ≡ w > −∞.

For every 1 ≤ i ≤ bnpc+ t− 1, define

I ′1(i) ≡

{I ∈ I :

max({i′ ∈ I : i′ < bnpc+ t}) = i, and

min({i′ ∈ I : i′ ≥ bnpc+ t}) ≥ bnpc+ t̄

},

and define

I ′1(0) ≡

{I ∈ I :

{i′ ∈ I : i′ < bnpc+ t}) = ∅, and

min({i′ ∈ I : i′ ≥ bnpc+ t}) ≥ bnpc+ t̄

}.

By construction, we have (cf. (52) for the definition of I ′1):

C(I ′1) =⋃

0≤i≤bnpc+t−1

C(I ′1(i)) (59)

For any 0 ≤ i ≤ bnpc+ t− 1 we have

Pn(C(I ′1(i))) ≤ Pn

(b∫ wi+1

wiG(x) dx ≥ F (mi+1)

∫ wbnpc+t̄

wi+1(1−G(x)) dx,

b∫ mi+1

miF (x) dx ≥ G(wi+1)

∫ mbnpc+t̄

mi+1(1− F (x)) dx

)(60)

because types in C(I ′1(i)) are such that given agents of rank i wait to the second period

while the next rank above i who waits is at least bnpc + t̄, some woman of rank j ≥ i + 1

must have incentive to match early with the man of rank i + 1, and some man of rank

j′ ≥ i+1 must have incentive to match early with the woman of rank i+1. Let’s look at the

incentive of the woman of rank j: her “downside” from waiting must dominate her “upside”

(see equation (56) on page 47); she has an “downside” of at most

(1−G(wj))

∫ mi+1

mi

V ′(x | wj)F (x) dx ≤∫ mi+1

mi

V ′(x | wj)F (x) dx ,

and an “upside” of at least

G(wj)

∫ mbnpc+t̄

mi+1

V ′(x | wj)(1− F (x)) dx ≥ G(wi+1)

∫ mbnpc+t̄

mi+1

V ′(x | wj)(1− F (x)) dx ;

and likewise for man of rank j′. Finally, for any x and y we have V ′(x | wj)/V ′(y | wj) ≤ b

by assumption (4). This explains the inequalities inside of the probability in (60).

50

Fix a ε > 0 such that p+ ε < 1, p− 3ε > 0 and t̄/n < ε/2.

Equations (59) and (60) imply that:

Pn(C(I ′1))

≤bnpc+t∑i=1

Pn

(b∫ wi

wi−1G(x) dx ≥ F (mi)

∫ wbnpc+t̄

wi(1−G(x)) dx,

b∫ mi

mi−1F (x) dx ≥ G(wi)

∫ mbnpc+t̄

mi(1− F (x)) dx

)

≤bnpc+t∑

i=bnp0c+1

Pn

(b∫ wi

wi−1G(x) dx ≥ F (mi)

∫ wbnpc+t̄

wi(1−G(x)) dx,

b∫ mi

mi−1F (x) dx ≥ G(wi)

∫ mbnpc+t̄

mi(1− F (x)) dx

)

+

bnp0c∑i=1

Pn

(b∫ wi

wi−1G(x) dx ≥ F (mi)

∫ wbnpc+t̄

wi(1−G(x)) dx,

b∫ mi

mi−1F (x) dx ≥ G(wi)

∫ mbnpc+t̄

mi(1− F (x)) dx

)

≤bnpc+t∑

i=bnp0c+1

Pn

(b(wi − wi−1) ≥ F (mbnp0c)(1−G(wbnpc+t̄))(wbnpc+t̄ − wi),b(mi −mi−1) ≥ G(wbnp0c)(1− F (mbnpc+t̄))(mbnpc+t̄ −mi)

)(61)

+

bnp0c∑i=1

Pn

(bG(wi)(wi − wi−1) ≥ F (mi)(1−G(wbnpc+t̄))(wbnpc+t̄ − wbnp0c),

bF (mi)(mi −mi−1) ≥ G(wi)(1− F (mbnpc+t̄))(mbnpc+t̄ −mbnp0c)

),

(62)

where p0 ∈ (ε, p− 2ε) is arbitrary.

Our goal is then to bound (61) and (62).

Choose a > a > 0 so that

a ≤ f(x), g(y) ≤ a,

for all (x, y) ∈ [F−1(p0 − ε), F−1(p+ ε)]× [G−1(p0 − ε), G−1(p+ ε)]. For (61) we have:

51

Pn

(b(wi − wi−1) ≥ F (mbnp0c)(1−G(wbnpc+t̄))(wbnpc+t̄ − wi),b(mi −mi−1) ≥ G(wbnp0c)(1− F (mbnpc+t̄))(mbnpc+t̄ −mi)

)

≤Pn

b(wi − wi−1) ≥ F (mbnp0c)(1−G(wbnpc+t̄))(wbnpc+t̄ − wi),G(wbnpc+t̄) ≤ p+ ε, F (mbnp0c) ≥ p0 − ε,b(mi −mi−1) ≥ G(wbnp0c)(1− F (mbnp0c+t̄))(mbnpc+t̄ −mi),

F (mbnpc+t̄) ≤ p+ ε, G(wbnp0c) ≥ p0 − ε

+ Pn(G(wbnpc+t̄) > p+ ε) + Pn(F (mbnpc+t̄) > p+ ε)

+ Pn(F (mbnp0c) < p0 − ε) + Pn(G(wbnp0c) < p0 − ε)

≤Pn

((a/a)b(vi − vi−1) ≥ (p0 − ε)(1− p− ε)(vbnpc+t̄ − vi),(a/a)b(ui − ui−1) ≥ (p0 − ε)(1− p− ε)(ubnpc+t̄ − ui)

)+ Pn(G(wbnpc+t̄) > p+ ε) + Pn(F (mbnpc+t̄) > p+ ε)

+ Pn(F (mbnp0c) < p0 − ε) + Pn(G(wbnp0c) < p0 − ε),

where it is w.l.o.g. to set mi ≡ F−1(ui) and wi ≡ G−1(vi): u1 ≤ . . . ≤ un and v1 ≤ . . . ≤ vn

are (independent copies of) order statistics of n i.i.d. U[0, 1] random variables (let u0 ≡ 0

and v0 ≡ 0).

By construction, we have 0 < a ≤ f(m), g(w) ≤ a <∞ for all m ∈ [F−1(p0− ε), F−1(p+

ε)] and w ∈ [G−1(p0 − ε), G−1(p+ ε)]. Thus, by the mean value theorem, we have for i > j:

(ui − uj)/a ≤ mi −mj ≤ (ui − uj)/a

and

(vi − vj)/a ≤ wi − wj ≤ (vi − vj)/a,

which explains the presence of a/a in the above probabilities.

By Lemma 3, we have

limn→∞

n

(Pn(G(wbnpc+t̄) > p+ ε) + Pn(F (mbnpc+t̄) > p+ ε)

+ Pn(F (mbnp0c) < p0 − ε) + Pn(G(wbnp0c) < p0 − ε))

= 0.

52

Therefore, to bound lim supn→∞ Pn(C(I ′1)) we can replace (61) by

bnpc+t∑i=bnp0c+1

Pn

((a/a)b(vi − vi−1) ≥ (p0 − ε)(1− p− ε)(vbnpc+t̄ − vi),(a/a)b(ui − ui−1) ≥ (p0 − ε)(1− p− ε)(ubnpc+t̄ − ui)

). (63)

Similarly, we can replace (62) by

bnp0c∑i=1

Pn

(abG(wi)(wi − wi−1) ≥ F (mi)(1− p− ε)(p− p0 − 2ε),

abF (mi)(mi −mi−1) ≥ G(wi)(1− p− ε)(p− p0 − 2ε)

). (64)

The following lemma takes care of (64):

Lemma 7.

limn→∞

bnp0c∑i=1

Pn

(abG(wi)(wi − wi−1) ≥ F (mi)(1− p− ε)(p− p0 − 2ε),

abF (mi)(mi −mi−1) ≥ G(wi)(1− p− ε)(p− p0 − 2ε)

)= 0.

Proof. For any 1 ≤ i ≤ bnp0c, we have:

Pn

(abG(wi)(wi − wi−1) ≥ F (mi)(1− p− ε)(p− p0 − 2ε),

abF (mi)(mi −mi−1) ≥ G(wi)(1− p− ε)(p− p0 − 2ε)

)

≤Pn

((wi − wi−1)(mi −mi−1) ≥

((1− p− ε)(p− p0 − 2ε)

ab

)2)

≤Pn(

(mi −mi−1) ≥ (1− p− ε)(p− p0 − 2ε)

ab

)+ Pn

((wi − wi−1) ≥ (1− p− ε)(p− p0 − 2ε)

ab

).

Let

C ≡ (1− p− ε)(p− p0 − 2ε)

ab,

and

δ ≡ supx∈[m,F−1(p0+ε)]

1− F (x+ C)

1− F (x)< 1.

We have

Pn((mi −mi−1) ≥ C) ≤ Pn((mi −mi−1) ≥ C,F (mi−1) ≤ p0 + ε) + 2 exp(−2ε2n)

≤ δn−bnp0c + 2 exp(−2ε2n),

53

for every 1 ≤ i ≤ bnp0c.And likewise for Pn((wi − wi−1) ≥ C).

By Lemma 4 and 5, we have:

bnpc+t∑i=bnp0c+1

Pn

((a/a)b(vi − vi−1) ≥ (p0 − ε)(1− p− ε)(vbnpc+t̄ − vi),(a/a)b(ui − ui−1) ≥ (p0 − ε)(1− p− ε)(ubnpc+t̄ − ui)

)

≤bnpc+t∑

i=bnp0c+1

(1 +

a(p0 − ε)(1− p− ε)ab

)−(bnpc+t̄−i)(1 +

a(p0 − ε)(1− p− ε)ab

)−(bnpc+t̄−i)

≤

(1−

(1 +

a(p0 − ε)(1− p− ε)ab

)−2)(

1 +a(p0 − ε)(1− p− ε)

ab

)−2s

,

since t̄− t = s+ 1.

Therefore, we have

lim supn→∞

Pn(C(I ′1)) ≤

(1−

(1 +

a(p0 − ε)(1− p− ε)ab

)−2)(

1 +a(p0 − ε)(1− p− ε)

ab

)−2s

.

(65)

By exact same argument, we have

lim supn→∞

Pn(C(I ′2)) ≤

(1−

(1 +

a(p0 − ε)(1− p− ε)ab

)−2)(

1 +a(p0 − ε)(1− p− ε)

ab

)−2s

.

(66)

These are the required bounds for Equation (53) and complete the proof of Theorem 2.

B Detail for Example 5

Consider the setting of Example 5, and let L = {m1,m2} ∪ {w1, w2}, i.e., all first-period

agents go to period 2.

As in Example 2 (Inequality (23)), one can show that with fixed transfers, man m1 has

an incentive to match early with woman w2, i.e., m1w2/2 > U(L,m1) if and only if

m1

2· (w2 − w1) +

m1

2· (1−m1)(w1)2 >

m1

2·m1((1− w1)2 − (1− w2)2), (67)

where the differences with Inequality (23) are that we do not divide out m1

2, and that we

have the correction term m1

2(w2 − w1) on the left-hand side because m1 is thinking about

54

matching early with w2 instead of w1. Inequality (67) always holds (because w2 − w1 >

(1 − w1)2 − (1 − w2)2), which reflects the fact that m1 is always matched to a type lower

than w2 in period 2 (since k = 1), so he has an incentive to match early with w2.

Likewise, with fixed transfers woman w2 has an incentive to match early with man m1,

i.e., m1w2/2 > V(L,w2) if and only if

w2

2· (m1 −m2) +

w2

2· (1− w2)((m2)2 − (m1)2) >

w2

2· w2(1−m2)2. (68)

Inequality (68) can never hold (because m2 −m1 > (m2)2 − (m1)2), which reflects the fact

that w2 is always matched to a type higher than m1 in period 2, so without transfers she

does not have an incentive to match early with m1.

Adding (67) and (68) together, we see that m1 and w2 have an incentive to match early

given some transfers, i.e., m1w2 > U(L,m1) + V(L,w2), if and only if

m1

2· (w2 − w1) +

m1

2· (1−m1)(w1)2 +

w2

2· (m1 −m2) +

w2

2· (1− w2)((m2)2 − (m1)2)

>m1

2·m1((1− w1)2 − (1− w2)2) +

w2

2· w2(1−m2)2 (69)

Finally, it is trivial to check that (69) holds for the realization of Example 5 in (47).

C Monte Carlo Simulation

We assume F = G = U[0, 1] and U(w | m) = V (m | w) = mw/2. To compute the

probability of sequential stability for fixed values of n and k, we make repeated draws of

the 2n agents’ types in the first period (from the U[0, 1] distribution), and for each draw we

enumerate all possible early matching scheme to check if one is sequentially stable in the

sense of Definition 2. To check sequential stability, we need an explicit formula of an agent’

expected utility in the second period, given a list of first-period agents who are waiting for

the second period. We now normalize U(w | m) = mw/2 to w and derive this formula from

the man’s perspective. This formula should be useful for future research when k is large.

Without loss of generality, let L = {mi : 1 ≤ i ≤ r} ∪ {wi : 1 ≤ i ≤ r} be a ranked19 list

of first-period men and women who wait for the second period. Man mi’s expected utility

19That is, we have m1 ≤ . . . ≤ mr and w1 ≤ . . . ≤ wr.

55

in the second period is:

U(L,mi) ≡k∑j=0

(k

j

)(mi)

j(1−mi)k−j · Ek[w2

i+j ],

where w2i+j is the (i + j)-th lowest woman in the second period, among the k new arrivals

and w1 ≤ . . . ≤ wr.

An explicit formula for E[w2s ], where s ≤ k + r, can be obtained as follows:

Ek[w2s ] =

∫ 1

0

Pk(w2s ≥ x) dx

=

∫ w1

0

s−1∑l=0

(k

l

)xl(1− x)k−l dx

+

∫ w2

w1

s−2∑l=0

(k

l

)xl(1− x)k−l dx

+ . . .

+

∫ ws

ws−1

(1− x)k dx

=s∑l=1

∫ wl

0

(k

s− l

)xs−l(1− x)k−(s−l) dx

=1

k + 1

s∑l=1

(1−B(s− l | wl, k + 1)), (70)

where if s > r, we set wr+1 = . . . = ws = 1 (which does not effect the value of w2s because

s ≤ k + r), and B( · | p, n) is the CDF of a binomial distribution with n independent trials

and probability p of success in each trial. Note that if w1 = w2 = . . . = ws = 1, then

Equation (70) gives Ek[w2s ] = s/(k + 1), clearly the right answer.

56